Constructing spoke subfactors using the jellyfish algorithm

This paper relies on some substantial calculations. In particular, our efforts to find the generators in the various graph planar algebras made use of a variety of techniques, some ad-hoc, some approximate, and some computationally expensive. This paper essentially does not address that work. Instead, we merely present the discovered generators and verify some relatively easy facts about them. In particular, the proofs presented in this paper rely on the computer in a much weaker sense. We need to calculate certain numbers of the form $\operatorname{Tr}(STR)$, where $S$, $T$, and $R$ are rather large martrices, and the computer does this for us. We also entered all the formulas derived in this paper into Mathematica, and had the computer automatically evaluate the various quantities which appear in our derivation of jellyfish relations. As a reader may be interested in seeing these programs, we include a brief instruction on finding and running these programs.

The arXiv sources of this article contain in the code subdirectory a number of files, including:

• •

  Generators.nb, which reconstructs the generators from our terse descriptions of them in Appendix A.

• •

  QuadraticTangles.nb, which calculates the requisite moments of these generators, and performs the linear algebra necessary to derive the jellyfish relations.

• •

  GenerateLaTeX.nb, which typesets each subsection of Section 4 for each planar algebra, and many mathematical expressions in Appendices A and B.

The Mathematica notebook Generators.nb can be run by itself. The final cells of that notebook write the full generators to the disk; this must be done before running QuadraticTangles.nb. The QuadraticTangles.nb notebook relies on the FusionAtlas, a substantial body of code the authors have developed along with Emily Peters, Noah Snyder and James Tener to perform calculations with subfactors and fusion categories. To obtain a local copy, you first need to ensure that you have Mercurial, the distributed version control system, installed on your machine. With that, the command

hg clone https://bitbucket.org/fusionatlas/fusionatlas

will create a local directory called fusionatlas containing the latest version. In the QuadraticTangles.nb notebook, you will then need to adjust the paths appearing in the first input cell to ensure that your local copy is included. After that, running the entire notebook reproduces all the calculations described below.

We invite any interested readers to contact us with questions or queries about the use of these notebooks or the FusionAtlas package.

A finite depth subfactor planar algebra embeds in the graph planar algebra of its principal graph [JP11, MW]. We begin by assuming that the desired planar algebra is $n-1$ supertransitive, so $P_{k,+}=TL_{k,+}$ for $k=0,\ldots,n-1$ and is generated by the orthogonal complement $P_{n,+}\ominus TL_{n,+}$ of the Temperley-Lieb algebra in the $n$-box space. If this orthogonal complement has dimension $k$, we say that the planar algebra is $n-1$ supertransitive of excess $k$. The excess can be read off from the principal graph; it’s the sum of the squares of the multiplicities of the edges immediately beyond the branch point, minus one (or simply two less than the valence of the branch point, in the simply laced case).

To construct the subfactor planar algebras in these papers, we first find a connection on the desired pair of principal graphs $(\Gamma,\Gamma^{\prime})$. We do so by first computing a numerical approximation to high precision, then guessing the exact solution using Mathematica’s RootApproximant function, and finally verifying using exact arithmetic in a number field that the candidate exact solution satisfies the necessary equations. As we use this numerical method, we cannot say with certainty how many connections exist on these graphs. However, Izumi has determined all such connections, and we believe our list of possible connections coincides with his. As the main goal of this paper is to construct the new 4442 subfactor by the development of methods for deriving jellyfish relations, rather than proving uniqueness, we restrict our attention to a single connection.

Using this connection, it is possible to discover the rotational eigenvalues of the desired generators via [IJMS11, Theorem 1.7]; in a certain gauge, $\text{tr}(UU^{t})=2+\sum\omega$, where $U$ is the ‘branch matrix’ of connection entries going through the branch point, and the sum is over the $k$ rotational eigenvalues $\omega$, with multiplicity, of the lowest weight vectors in the $n$-box space. We then compute the equations for flatness and find $k$ flat $n$-boxes with respect to the connection. (Note that these flat elements are not necessarily self-adjoint as in [Jon03].)

However, merely having flat elements for a connection does not imply that the connection itself is flat. The planar algebra generated by the flat $n$-boxes is necessarily evaluable, and indeed a subfactor planar algebra, which we denote by $P_{\bullet}^{\Gamma}$. Thus we have constructed some subfactor by [Ocn88, EK98, Jon11, MW]. This subfactor, however, need not have the principal graphs we started with; indeed, it could have smaller supertransitivity. Determining even the supertransitivity of the principal graph of the subfactor planar algebra generated by the flat $n$-boxes requires some knowledge of the skein theory of the planar algebra, which is of comparable difficulty with directly computing jellyfish relations. In fact, by Lemma 5.1, having jellyfish relations ensures the supertransitivity is correct. One could attempt to verify the connection itself is flat, which would also ensure the supertransitivity is correct, but this is extremely computationally expensive. The point of this paper is to avoid this work using planar algebras and the jellyfish algorithm.

Hence, we still need to analyze the subfactor planar algebra $P_{\bullet}^{\Gamma}$. In the analysis which follows, we never rely on the fact that we obtained these elements as flat elements with respect to some connection. In particular, nothing that follows relies on the correctness of the method described above for guessing the generators!

First, we need to calculate the principal graphs of $P_{\bullet}^{\Gamma}$ (which we are hoping are the graphs we started with, $(\Gamma,\Gamma^{\prime})$). Second, we would like to know how to evaluate closed diagrams in the generators.

For the spoke graphs we consider in this paper, $\Gamma$ and $\Gamma^{\prime}$ always coincide, and are drawn from the set $\{4442,3333,3311,2221\}$ (see Figure 1), where the numbers refer to the length of the spokes emanating from the central vertex. By [BP12], if such subfactor planar algebras were to exist with the desired principal graphs, they would have 1-strand jellyfish generators at depth $n$. We use Jones’ quadratic tangles techniques [Jon03] to find these 1-strand jellyfish relations in $P_{\bullet}^{\Gamma}$. We then use the generators to determine that $P_{\bullet}^{\Gamma}$ has the correct principal graph.

In [Jon03], Jones uses quadratic tangles techniques to prove a number of formulas about planar generators in a subfactor planar algebra. In fact, many of the formulas there hold in more generality.

In [MP12], Morrison and Peters describe the spherical and lopsided conventions for planar algebras. In the spherical convention, both shaded and unshaded contractible closed loops count for a multiplicative factor of $\delta$; however, in the lopsided convention, shaded contractible closed loops count for $1$ while unshaded contractible closed loops count for $\delta^{2}$. The main advantage of working in the lopsided planar algebra is that there are fewer square roots, so arithmetic is easier. In particular the number field in which we calculate is much smaller. Hence we use the lopsided planar algebra to compute the moments of our generators.

The map $\natural\colon P_{\bullet}^{spherical}\to P_{\bullet}^{lopsided}$ from [MP12] is not a planar algebra map, but it commutes with the action of the planar operad up to certain scalars. When we draw our tangles in the standard form where each input and output disk is a rectangle with the distinguished interval on the left and the same number of strings attach to the top and bottom of each rectangle, then there is a power of $\delta^{\pm 1}$ for each critical point which is shaded above, and the power of $\delta$ corresponds to the sign of the critical point:

$$\leavevmode\hbox to28.45pt{\vbox to21.54pt{\pgfpicture\makeatletter\hbox{\qquad\lower-14.22638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\pgfsys@moveto{-14.22638pt}{-14.22638pt}\pgfsys@lineto{-14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{}{{}}{} {}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{0.826,0.82,0.874}\pgfsys@color@rgb@fill{0.826}{0.82}{0.874}\pgfsys@invoke{ }{}\pgfsys@moveto{7.11319pt}{7.11319pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{-3.92854pt}{3.92854pt}{-7.11319pt}{0.0pt}{-7.11319pt}\pgfsys@curveto{-3.92854pt}{-7.11319pt}{-7.11319pt}{-3.92854pt}{-7.11319pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\longleftrightarrow\delta\hskip 42.67912pt\leavevmode\hbox to28.45pt{\vbox to21.34pt{\pgfpicture\makeatletter\hbox{\qquad\lower-7.11319pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{} {}{}\pgfsys@moveto{-14.22638pt}{-7.11319pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@lineto{14.22638pt}{-7.11319pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{}{{}}{} {}{} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{0.826,0.82,0.874}\pgfsys@color@rgb@fill{0.826}{0.82}{0.874}\pgfsys@invoke{ }\pgfsys@moveto{-14.22638pt}{-14.22638pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{}{{}}{} {}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-7.11319pt}{-14.22638pt}\pgfsys@lineto{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{3.92854pt}{-3.92854pt}{7.11319pt}{0.0pt}{7.11319pt}\pgfsys@curveto{3.92854pt}{7.11319pt}{7.11319pt}{3.92854pt}{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\longleftrightarrow\delta^{-1}.$$

The jellyfish algorithm was invented in [BMPS12] to construct the extended Haagerup subfactor planar algebra with principal graphs

$$\left({\hskip-3.0pt\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[height=17.07164pt]{diagrams/graphs/C53E32050C862F37}}\end{array}\hskip-3.0pt},{\hskip-3.0pt\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[height=17.07164pt]{diagrams/graphs/CA1952BED6C8EB18}}\end{array}\hskip-3.0pt}\right).$$

One uses the jellyfish algorithm to evaluate closed diagrams on a set of generators. There are two ingredients:

1. (1)

   The generators in $P_{n,\pm}$ satisfy jellyfish relations, i.e., for each generator $S,T$,

   $$j(\check{S})=\leavevmode\hbox to40.23pt{\vbox to43.08pt{\pgfpicture\makeatletter\hbox{\hskip 20.11684pt\lower-22.96228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-19.91684pt}{-22.76228pt}\pgfsys@lineto{-19.91684pt}{0.0pt}\pgfsys@curveto{-19.91684pt}{10.99995pt}{-10.99986pt}{19.91693pt}{0.00009pt}{19.91693pt}\pgfsys@curveto{11.00005pt}{19.91693pt}{19.91702pt}{10.99995pt}{19.91702pt}{0.0pt}\pgfsys@lineto{19.91684pt}{-22.76228pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.22574pt}{-19.32738pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.14236pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\check{S}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.61319pt}{-16.55276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\,,\,j^{2}(S^{\prime})=\leavevmode\hbox to51.61pt{\vbox to48.77pt{\pgfpicture\makeatletter\hbox{\hskip 25.8073pt\lower-22.96228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-25.6073pt}{-22.76228pt}\pgfsys@lineto{-25.6073pt}{0.0pt}\pgfsys@curveto{-25.6073pt}{14.14279pt}{-14.14261pt}{25.60748pt}{0.00018pt}{25.60748pt}\pgfsys@curveto{14.14297pt}{25.60748pt}{25.60767pt}{14.14279pt}{25.60767pt}{0.0pt}\pgfsys@lineto{25.6073pt}{-22.76228pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{}{{}}{} {}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-19.91684pt}{-22.76228pt}\pgfsys@lineto{-19.91684pt}{0.0pt}\pgfsys@curveto{-19.91684pt}{10.99995pt}{-10.99986pt}{19.91693pt}{0.00009pt}{19.91693pt}\pgfsys@curveto{11.00005pt}{19.91693pt}{19.91702pt}{10.99995pt}{19.91702pt}{0.0pt}\pgfsys@lineto{19.91684pt}{-22.76228pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.22574pt}{-19.32738pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61632pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.61319pt}{-16.55276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}$$

   can be written as linear combinations of trains, which are diagrams where any region meeting the distinguished interval of a generator meets the distinguished interval of the external disk, e.g.,

   where $S_{1},\dots,S_{\ell}$ are generators, and ${\mathcal{T}}$ is a single Temperley-Lieb diagram.

   (Note that $j(S),j(\check{S})$ means the same thing as $\cup_{0}(S),\cup_{0}(\check{S})$, but we will use the $j$ notation to emphasize its importance to the jellyfish algorithm.)

2. (2)

   The generators in $P_{n,\pm}$ are uncappable and together with the Jones-Wenzl projection $f^{(n)}$ form an algebra under the usual multiplication

   $$ST=\leavevmode\hbox to33.26pt{\vbox to74.38pt{\pgfpicture\makeatletter\hbox{\hskip 21.4821pt\lower-11.58092pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{62.59598pt}\pgfsys@lineto{0.0pt}{-11.38092pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@moveto{11.3811pt}{42.67914pt}\pgfsys@curveto{11.3811pt}{48.96483pt}{6.28569pt}{54.06024pt}{0.0pt}{54.06024pt}\pgfsys@curveto{-6.28569pt}{54.06024pt}{-11.3811pt}{48.96483pt}{-11.3811pt}{42.67914pt}\pgfsys@curveto{-11.3811pt}{36.39345pt}{-6.28569pt}{31.29803pt}{0.0pt}{31.29803pt}\pgfsys@curveto{6.28569pt}{31.29803pt}{11.3811pt}{36.39345pt}{11.3811pt}{42.67914pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61632pt}{39.26248pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.1491pt}{40.35275pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ 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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.1491pt}{6.20952pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.16212pt}{24.10036pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.16212pt}{-7.1974pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.16212pt}{55.39857pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}=\sum_{R}\alpha^{R}_{S,T}\leavevmode\hbox to33.26pt{\vbox to45.92pt{\pgfpicture\makeatletter\hbox{\hskip 21.4821pt\lower-22.96228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{22.76228pt}\pgfsys@lineto{0.0pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.3811pt}{0.0pt}\pgfsys@curveto{11.3811pt}{6.28569pt}{6.28569pt}{11.3811pt}{0.0pt}{11.3811pt}\pgfsys@curveto{-6.28569pt}{11.3811pt}{-11.3811pt}{6.28569pt}{-11.3811pt}{0.0pt}\pgfsys@curveto{-11.3811pt}{-6.28569pt}{-6.28569pt}{-11.3811pt}{0.0pt}{-11.3811pt}\pgfsys@curveto{6.28569pt}{-11.3811pt}{11.3811pt}{-6.28569pt}{11.3811pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.83507pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$R$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.1491pt}{-2.32639pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.16212pt}{15.56488pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.16212pt}{-18.57877pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\,.$$

   (Note that the Mathematica package FusionAtlas multiplies in this order; reading from left to right in products corresponds to reading from bottom to top in planar composites.)

Given these two ingredients, one can evaluate any closed diagram using the following two step process.

1. (1)

   Pull all generators $S$ to the outside of the diagram using the jellyfish relations, possibly getting diagrams with more $S$’s.

2. (2)

   Use uncappability and the algebra property to iteratively reduce the number of generators. Any non-zero train which is a closed diagram is either a Temperley-Lieb diagram, has a capped generator, or has two generators $S,T$ connected by at least $n$ strings, giving $ST$.

Section 3 is devoted to our procedure for computing the jellyfish relations necessary for the first part of the jellyfish algorithm, while the second part is rather easy.

One can see that if $\{A,B,f^{(n)}\}$ span a subalgebra of $P_{n,+}$, their structure coefficients must be given by

$$\alpha_{S,T}^{R}=\frac{\operatorname{Tr}(STR)}{\operatorname{Tr}(R^{2})},$$

and thus determined by the moments given in Appendix B. We check that the algebra generated by $\{A,B,f^{(n)}\}$ is closed under multiplication directly in the graph planar algebra, in Lemma 4.1.

Note that if we have an orthonormal set of generators, then in the notation of [Jon03], $\alpha_{S,T}^{R}=\operatorname{Tr}(STR)=a^{ST}_{R}$. There are similar easy calculations to determine the structure coefficients $\beta_{S,T}^{R}$ of the algebra generated by $\{\check{A},\check{B},\check{f}^{(n)}\}\subset P_{n,-}$, and in the orthonormal case, $\beta_{S,T}^{R}=b^{ST}_{R}$.

In [BMPS12], they found 2-strand jellyfish relations, i.e., $j(\check{S})$ and $j^{2}(S)$ lie in the span of the trains of $S$. For spoke subfactors, we can find 1-strand jellyfish relations by [BP12], i.e., for each $S\in{\mathfrak{B}}$ and $\check{S}\in\check{{\mathfrak{B}}}$, $j(\check{S})$ lies in the span of the trains from ${\mathfrak{B}}$, and $j(S)$ lies in the span of the trains from $\check{{\mathfrak{B}}}$. These 1-strand jellyfish relations are sufficient to evaluate all closed diagrams from our generators, and thus ${\mathfrak{B}}$ generates some subfactor planar algebra.

We need further arguments to prove that the resulting subfactor planar algebra has the desired principal graphs. It turns out that at the relatively low index of $3+\sqrt{5}$ these arguments are easy.

Given a set of jellyfish relations, we may write them in a more compact form in which we multiply the diagrams by a Jones-Wenzl idempotent to get rid of simpler diagrams. We present the arguments back and forth in the case of one generator, and it is clear how to generalize to the multi-generator case.

If we know

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{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.22574pt}{-19.32738pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.14236pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\check{S}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} 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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@moveto{56.9055pt}{0.0pt}\pgfsys@curveto{56.9055pt}{6.28558pt}{51.81017pt}{11.38092pt}{45.52458pt}{11.38092pt}\pgfsys@curveto{39.239pt}{11.38092pt}{34.14366pt}{6.28558pt}{34.14366pt}{0.0pt}\pgfsys@curveto{34.14366pt}{-6.28558pt}{39.239pt}{-11.38092pt}{45.52458pt}{-11.38092pt}\pgfsys@curveto{51.81017pt}{-11.38092pt}{56.9055pt}{-6.28558pt}{56.9055pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{42.17043pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{43.02458pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}+\sum_{i=1}^{2n+1}\gamma_{S,i}\cup_{i}^{\prime}(S)+X$$

where $X\in TL_{n+1,+}$ with all strings turned down, and

$$\cup_{1}^{\prime}(S)=\leavevmode\hbox to30.76pt{\vbox to44.27pt{\pgfpicture\makeatletter\hbox{\qquad\lower-22.96228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{}{}\pgfsys@moveto{-17.07182pt}{-22.76228pt}\pgfsys@curveto{-17.07182pt}{-19.61945pt}{-14.52411pt}{-17.07173pt}{-11.38127pt}{-17.07173pt}\pgfsys@curveto{-8.23843pt}{-17.07173pt}{-5.69072pt}{-19.61945pt}{-5.69072pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.22574pt}{-19.32738pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\,,\,\cup_{i}^{\prime}(S)=\leavevmode\hbox to92.97pt{\vbox to44.45pt{\pgfpicture\makeatletter\hbox{\hskip 35.9174pt\lower-23.13747pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{-5.69046pt}{0.0pt}\pgfsys@lineto{-17.07182pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{17.07182pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{}{}\pgfsys@moveto{-5.69046pt}{-22.76228pt}\pgfsys@curveto{-5.69046pt}{-19.61945pt}{-3.14275pt}{-17.07173pt}{0.00009pt}{-17.07173pt}\pgfsys@curveto{3.14293pt}{-17.07173pt}{5.69064pt}{-19.61945pt}{5.69064pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-32.5844pt}{-19.38779pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$i-1$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.2552pt}{-18.97112pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n-i+1$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\,,\text{ and }\cup_{2n+1}^{\prime}(S)=\leavevmode\hbox to30.76pt{\vbox to44.27pt{\pgfpicture\makeatletter\hbox{\qquad\lower-22.96228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{}{}\pgfsys@moveto{5.69046pt}{-22.76228pt}\pgfsys@curveto{5.69046pt}{-19.61945pt}{8.23817pt}{-17.07173pt}{11.38101pt}{-17.07173pt}\pgfsys@curveto{14.52385pt}{-17.07173pt}{17.07156pt}{-19.61945pt}{17.07156pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.15518pt}{-19.32738pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}$$

(note that the $\cup^{\prime}_{i}(S)$’s for $i=1,\dots,2n+1$ can be obtained from the $\cup_{j}(S)$’s for $j=1,\dots,2n+1$ by applying suitable powers of the rotation and multiplying by suitable powers of $\sigma_{S}$), applying the Jones-Wenzl $f^{(2n+2)}$ to the bottom of the diagram gives the simpler box jellyfish relation

$$f\cdot j(\check{S})=\leavevmode\hbox to67.41pt{\vbox to68.69pt{\pgfpicture\makeatletter\hbox{\hskip 38.5535pt\lower-48.5696pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-19.91684pt}{-48.3696pt}\pgfsys@lineto{-19.91684pt}{0.0pt}\pgfsys@curveto{-19.91684pt}{10.99995pt}{-10.99986pt}{19.91693pt}{0.00009pt}{19.91693pt}\pgfsys@curveto{11.00005pt}{19.91693pt}{19.91702pt}{10.99995pt}{19.91702pt}{0.0pt}\pgfsys@lineto{19.91684pt}{-48.3696pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-48.3696pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.22574pt}{-19.32738pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.14236pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\check{S}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.61319pt}{-16.55276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{} {}{} {}{} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-28.45276pt}{-42.67914pt}\pgfsys@lineto{-28.45276pt}{-22.76228pt}\pgfsys@lineto{28.45276pt}{-22.76228pt}\pgfsys@lineto{28.45276pt}{-42.67914pt}\pgfsys@lineto{-28.45276pt}{-42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.8884pt}{-36.18773pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$f^{(2n+2)}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.22049pt}{-35.04688pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}=\gamma_{S,S}\leavevmode\hbox to88.2pt{\vbox to68.29pt{\pgfpicture\makeatletter\hbox{\qquad\lower-48.3696pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{} {}{}\pgfsys@moveto{-19.91684pt}{-48.3696pt}\pgfsys@lineto{-19.91684pt}{19.91684pt}\pgfsys@lineto{68.28644pt}{19.91684pt}\pgfsys@lineto{68.28644pt}{-48.3696pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{}{{}}{} {}{} {}{} {}{}{}\pgfsys@moveto{0.0pt}{-56.90552pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.72816pt}{2.01218pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n-1$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.62457pt}{-18.9107pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n+1$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ 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} \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@moveto{56.9055pt}{0.0pt}\pgfsys@curveto{56.9055pt}{6.28558pt}{51.81017pt}{11.38092pt}{45.52458pt}{11.38092pt}\pgfsys@curveto{39.239pt}{11.38092pt}{34.14366pt}{6.28558pt}{34.14366pt}{0.0pt}\pgfsys@curveto{34.14366pt}{-6.28558pt}{39.239pt}{-11.38092pt}{45.52458pt}{-11.38092pt}\pgfsys@curveto{51.81017pt}{-11.38092pt}{56.9055pt}{-6.28558pt}{56.9055pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{42.17043pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{43.02458pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{} {}{} {}{} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-11.38092pt}{-42.67914pt}\pgfsys@lineto{-11.38092pt}{-22.76228pt}\pgfsys@lineto{56.90552pt}{-22.76228pt}\pgfsys@lineto{56.90552pt}{-42.67914pt}\pgfsys@lineto{-11.38092pt}{-42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.87389pt}{-36.18773pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$f^{(2n+2)}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.1491pt}{-35.04688pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}.$$

Conversely, given a box jellyfish relation of the above form, we get a jellyfish relation by expanding the Jones-Wenzl idempotents.

1. (1)

   First, note that the coefficient of the identity in the Jones-Wenzl idempotent $f^{(2n+2)}$ is 1. This term gives us a $j(\check{S})$ on the left hand side and an $S\circ S$ on the right hand side.

2. (2)

   On the left hand side, all other terms of the Jones-Wenzl either cap off the $\check{S}$, giving zero, or there is exactly one cup on the top and one cap on the bottom, and the cup on the top is between the $1$-st and $2$-nd strings or the $(2n+1)$-th and $(2n+2)$-th strings. In this case, we get a scalar multiple of an annular consequence of $S$ (which is not $j(\check{S})$), and these terms can be subtracted off to the right hand side of the equation.

3. (3)

   On the right hand side, there are a few more options. First, if any $S$ is capped off, we get zero. Otherwise there is a cup between the $(n+1)$-th and $(n+2)$-th strings, and we get an an $S^{2}$ which can be written as a linear combination of $S$ and Temperley-Lieb diagrams. Now if there are more cups on top of the term in the Jones-Wenzl, the $S$ term vanishes, and we get a Temperley-Lieb diagram. If there are no extra cups, then we are left with some Temperley-Lieb diagrams and some scalar multiple of an annular consequence of $S$ (which is not $j(\check{S})$).

Hence we get a jellyfish relation.

Given our set of generators ${\mathfrak{B}}=\{A,B\}\subset P_{n,+}$, we have the dual generators $\check{{\mathfrak{B}}}=\{\check{A},\check{B}\}\subset P_{n,-}$, and ${\mathfrak{B}}$ and $\check{{\mathfrak{B}}}$ each give 4 quadratic tangles:

	$\displaystyle\{A\circ A,A\circ B,B\circ A,B\circ B\}$	$\displaystyle\subset P_{n+1,+}$
and
	$\displaystyle\{\check{A}\circ\check{A},\check{A}\circ\check{B},\check{B}\circ\check{A},\check{B}\circ\check{B}\}$	$\displaystyle\subset P_{n+1,-},$

where

$$S\circ T=\leavevmode\hbox to82.51pt{\vbox to42.68pt{\pgfpicture\makeatletter\hbox{\qquad\lower-22.76228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{} {}{}\pgfsys@moveto{-14.22638pt}{-22.76228pt}\pgfsys@lineto{-14.22638pt}{19.91684pt}\pgfsys@lineto{68.28644pt}{19.91684pt}\pgfsys@lineto{68.28644pt}{-22.76228pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{}{{}}{} {}{} {}{} {}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{-28.45276pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ 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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@moveto{56.9055pt}{0.0pt}\pgfsys@curveto{56.9055pt}{6.28558pt}{51.81017pt}{11.38092pt}{45.52458pt}{11.38092pt}\pgfsys@curveto{39.239pt}{11.38092pt}{34.14366pt}{6.28558pt}{34.14366pt}{0.0pt}\pgfsys@curveto{34.14366pt}{-6.28558pt}{39.239pt}{-11.38092pt}{45.52458pt}{-11.38092pt}\pgfsys@curveto{51.81017pt}{-11.38092pt}{56.9055pt}{-6.28558pt}{56.9055pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{41.90826pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{43.02458pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}.$$

Since we expect these generators to give a subfactor planar algebra with the principal graph being the underlying graph of the graph planar algebra in which they were found, we hope that some linear combinations of these quadratic tangles lie in the space $\textgoth{A}(A,B,\emptyset)$ of annular consequences of $A,B$, and the empty diagram (the annular consequences of the empty diagram are the diagrams in Temperley-Lieb), and the same for the checked generators:

	$\displaystyle{\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$	$\displaystyle={\textgoth{A}}(A,B,\emptyset)\cap\operatorname{span}(\{A\circ A,A\circ B,B\circ A,B\circ B\})$
	$\displaystyle{\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}^{\vee}$	$\displaystyle={\textgoth{A}}(\check{A},\check{B},\emptyset)\cap\operatorname{span}(\{\check{A}\circ\check{A},\check{A}\circ\check{B},\check{B}\circ\check{A},\check{B}\circ\check{B}\})$

(${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$ stands for “quadratic tangles in annular consequences”).

We use formulas from [Jon03] to calculate bases for ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$ and ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}^{\vee}$. We first describe how to find a basis for ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$.

First, we calculate the $4\times 4$ matrix of inner products modulo the annular consequences of $A,B$:

$$\left(\langle[S\circ T-P_{{\textgoth{A}}(A,B)}(S\circ T)],P\circ Q\rangle\right)_{(S,T),(P,Q)\in{\mathfrak{B}}^{2}}.$$

The inner products are given by the following formulas, where the second comes from Proposition 4.4.2 of [Jon03]:

	$\displaystyle\langle S\circ T,P\circ Q\rangle$	$\displaystyle=\frac{1}{[n]}\operatorname{Tr}(QT)\operatorname{Tr}(SP)$
	$\displaystyle\langle P_{{\textgoth{A}}(A,B)}S\circ T,P\circ Q\rangle$	$\displaystyle=\sum_{R\in{\mathfrak{B}}}\frac{1}{W_{R}}\bigg{\{}\left(\overline{a_{R}^{ST}}a_{R}^{PQ}+\sigma_{T}\overline{\sigma_{S}\sigma_{Q}}\sigma_{P}\overline{b_{R}^{ST}}b_{R}^{PQ}\right)\left(\omega_{R}^{-1}+[2n+2]\right)$
		$\displaystyle+(-1)^{n+1}\sigma_{R}\left(\overline{\sigma_{Q}}\sigma_{P}\overline{a_{R}^{ST}}b_{R}^{PQ}+\sigma_{T}\overline{\sigma_{S}}\overline{b_{R}^{ST}}a_{R}^{PQ}\right)\left(2\omega_{R}^{-1}[n+1]\right)\bigg{\}},$

where $a_{R}^{ST}=\operatorname{Tr}(RST)$, $b_{R}^{ST}=\operatorname{Tr}(\check{R}\check{S}\check{T})$, and $W_{R}=q^{2n+2}+q^{-2n+2}-\omega_{R}-\omega_{R}^{-1}$. Here, $q>1$ such that $\delta=[2]=q+q^{-1}$.

Taking a basis for the nullspace of this matrix gives us a basis for ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$.

Now we want to write each basis element of ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$ or ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}^{\vee}$ as a linear combination of the $\cup_{i}(R),\cup_{i}(\check{R})$’s for $R\in{\mathfrak{B}},\check{R}\in\check{{\mathfrak{B}}}$ respectively. We describe the process for the basis elements of ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$, and the checked versions are again computed by passing to the dual as in Remark 3.6. Using Proposition $4.4.1.i$ in [Jon03], we have

(3.1)

$$P_{{\textgoth{A}}(A,B)}(S\circ T)=\sum_{R\in\mathfrak{B}}\sigma_{R}^{n}a_{R}^{ST}\widehat{\cup}_{n+1}(R)+\sigma_{T}^{-1}\sigma_{S}b_{R}^{ST}\widehat{\cup}_{0}(R).$$

We can express $\widehat{\cup}_{n+1}(R),\widehat{\cup}_{0}(R)$ (in the dual annular basis) in terms of the annular basis $\cup_{i}(R)$ for $R\in{\mathfrak{B}}$, using the formulas from Proposition 4.2.9 of [Jon03]:

	$\displaystyle\widehat{\cup}_{0}(R)$	$\displaystyle=\frac{1}{W_{R}}\left([2n+2]\cup_{0}(R)+((-\sigma)^{n+1}+(-\sigma)^{-n-1})[n+1]\cup_{n+1}(R)+X\right)$
	$\displaystyle\widehat{\cup}_{n+1}(R)$	$\displaystyle=\frac{1}{W_{R}}\left([2n+2]\cup_{n+1}(R)+((-\sigma)^{n+1}+(-\sigma)^{-n-1})[n+1]\cup_{0}(R)+Y\right)$

where $X,Y\in\overline{{\textgoth{A}}(R)}=\operatorname{span}\left\{\cup_{i}(R)\middle|i\neq 0,n+1\right\}$.

Putting it all together, we can express $v$ in our basis of ${\mathcal{QT}\hskip-3.61371pt\mathcal{AC}}$ as a linear combination of $\cup_{0}(A),\cup_{0}(B),\cup_{n+1}(A),\cup_{n+1}(B)$, plus another element in $\overline{{\textgoth{A}}(A,B,\emptyset)}=TL_{n+1,\pm}\oplus\operatorname{span}(\left\{\cup_{i}(R)\middle|R\in{\mathfrak{B}}\text{ and }i\neq 0,n+1\right\})$.

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}}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-17.97548pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}+\nu_{B}\leavevmode\hbox to40.06pt{\vbox to40.43pt{\pgfpicture\makeatletter\hbox{\qquad\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{11.38092pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{-5.69046pt}{0.0pt}\pgfsys@lineto{-11.38092pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{}{}\pgfsys@moveto{-5.69046pt}{-28.45276pt}\pgfsys@curveto{-5.69046pt}{-25.30992pt}{-3.14275pt}{-22.7622pt}{0.00009pt}{-22.7622pt}\pgfsys@curveto{3.14293pt}{-22.7622pt}{5.69064pt}{-25.30992pt}{5.69064pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.69804pt}{-15.73332pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.75471pt}{-15.73332pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.45952pt}{-3.75891pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{${B}^{\vee n}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-17.97548pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}+Z,$

where $Z\in\overline{{\textgoth{A}}(A,B,\emptyset)}$ and

$$S^{\vee n}=\begin{cases}S&\text{if $n$ is even}\\ \check{S}&\text{if $n$ is odd}\end{cases}\hskip 36.135pt\check{S}^{\vee n}=\begin{cases}\check{S}&\text{if $n$ is even}\\ S&\text{if $n$ is odd.}\end{cases}$$

Given such an equation, we can multiply by a Jones-Wenzl idempotent in two ways to find the following relations, from which we will derive the desired box jellyfish relations:

1. (1)

   We can multiply by $f^{(2n+2)}$ on the bottom to isolate the $\cup_{0}(A),\cup_{0}(B)$:

   $$\sum_{S,T\in{\mathfrak{B}}}\gamma_{S,T}\leavevmode\hbox to88.2pt{\vbox to68.29pt{\pgfpicture\makeatletter\hbox{\qquad\lower-48.3696pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{} {}{}\pgfsys@moveto{-19.91684pt}{-48.3696pt}\pgfsys@lineto{-19.91684pt}{19.91684pt}\pgfsys@lineto{68.28644pt}{19.91684pt}\pgfsys@lineto{68.28644pt}{-48.3696pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{}{{}}{} {}{} {}{} {}{}{}\pgfsys@moveto{0.0pt}{-56.90552pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ 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} \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{43.02458pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{} {}{} {}{} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-11.38092pt}{-42.67914pt}\pgfsys@lineto{-11.38092pt}{-22.76228pt}\pgfsys@lineto{56.90552pt}{-22.76228pt}\pgfsys@lineto{56.90552pt}{-42.67914pt}\pgfsys@lineto{-11.38092pt}{-42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.87389pt}{-36.18773pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$f^{(2n+2)}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { 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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\check{A}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.61319pt}{-16.55276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{} {}{} {}{} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-28.45276pt}{-42.67914pt}\pgfsys@lineto{-28.45276pt}{-22.76228pt}\pgfsys@lineto{28.45276pt}{-22.76228pt}\pgfsys@lineto{28.45276pt}{-42.67914pt}\pgfsys@lineto{-28.45276pt}{-42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.8884pt}{-36.18773pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$f^{(2n+2)}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.22049pt}{-35.04688pt}\pgfsys@invoke{ 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}{}\pgfsys@moveto{-19.91684pt}{-48.3696pt}\pgfsys@lineto{-19.91684pt}{0.0pt}\pgfsys@curveto{-19.91684pt}{10.99995pt}{-10.99986pt}{19.91693pt}{0.00009pt}{19.91693pt}\pgfsys@curveto{11.00005pt}{19.91693pt}{19.91702pt}{10.99995pt}{19.91702pt}{0.0pt}\pgfsys@lineto{19.91684pt}{-48.3696pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-48.3696pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.22574pt}{-19.32738pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$2n$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ 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}\hbox{{$\check{B}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.61319pt}{-16.55276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{} {}{} {}{} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{-28.45276pt}{-42.67914pt}\pgfsys@lineto{-28.45276pt}{-22.76228pt}\pgfsys@lineto{28.45276pt}{-22.76228pt}\pgfsys@lineto{28.45276pt}{-42.67914pt}\pgfsys@lineto{-28.45276pt}{-42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.8884pt}{-36.18773pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$f^{(2n+2)}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.22049pt}{-35.04688pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}$$

   as any cap on top of $f^{(2n+2)}$ gives zero.

2. (2)

   We can bend $f^{(2n+2)}$ around the top to isolate the $\cup_{n+1}(A),\cup_{n+1}(B)$:

   $$\sum_{S,T\in{\mathfrak{B}}}\gamma_{S,T}\leavevmode\hbox to130.88pt{\vbox to91.05pt{\pgfpicture\makeatletter\hbox{\hskip 45.52458pt\lower-45.52458pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{} {}{}\pgfsys@moveto{-45.52458pt}{-45.52458pt}\pgfsys@lineto{-45.52458pt}{45.52458pt}\pgfsys@lineto{85.35828pt}{45.52458pt}\pgfsys@lineto{85.35828pt}{-45.52458pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{}{{}}{} {}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{-45.52458pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{-45.52458pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.72816pt}{2.01218pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n-1$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.62457pt}{-16.06526pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n+1$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.38687pt}{-18.9107pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptsize{$n+1$}}}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{11.38092pt}{0.0pt}\pgfsys@curveto{11.38092pt}{6.28558pt}{6.28558pt}{11.38092pt}{0.0pt}{11.38092pt}\pgfsys@curveto{-6.28558pt}{11.38092pt}{-11.38092pt}{6.28558pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-6.28558pt}{-6.28558pt}{-11.38092pt}{0.0pt}{-11.38092pt}\pgfsys@curveto{6.28558pt}{-11.38092pt}{11.38092pt}{-6.28558pt}{11.38092pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@moveto{56.9055pt}{0.0pt}\pgfsys@curveto{56.9055pt}{6.28558pt}{51.81017pt}{11.38092pt}{45.52458pt}{11.38092pt}\pgfsys@curveto{39.239pt}{11.38092pt}{34.14366pt}{6.28558pt}{34.14366pt}{0.0pt}\pgfsys@curveto{34.14366pt}{-6.28558pt}{39.239pt}{-11.38092pt}{45.52458pt}{-11.38092pt}\pgfsys@curveto{51.81017pt}{-11.38092pt}{56.9055pt}{-6.28558pt}{56.9055pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{41.90826pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{43.02458pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{13.32271pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\star$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} {}{{}}{}{{{}{}}}{}{{{}{}}}{}{}{} {}{}{{{}{}}}{}{{{}{}}}{}{}{} {}{}{{{}{}}}{}{{{}{}}}{}{}{} {}{}{{{}{}}}{}{{{}{}}}{}{}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{39.83368pt}{-22.76228pt}\pgfsys@curveto{73.97699pt}{-22.76228pt}{73.97699pt}{22.76228pt}{39.83368pt}{22.76228pt}\pgfsys@lineto{5.69046pt}{22.76228pt}\pgfsys@curveto{-28.45285pt}{22.76228pt}{-28.45285pt}{-22.76228pt}{5.69046pt}{-22.76228pt}\pgfsys@lineto{5.69046pt}{-39.83368pt}\pgfsys@curveto{-51.21506pt}{-39.83368pt}{-51.21506pt}{39.83368pt}{5.69046pt}{39.83368pt}\pgfsys@lineto{39.83368pt}{39.83368pt}\pgfsys@curveto{96.7392pt}{39.83368pt}{96.7392pt}{-39.83368pt}{39.83368pt}{-39.83368pt}\pgfsys@lineto{39.83368pt}{-22.76228pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{ }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.87389pt}{27.83096pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ 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   which is equivalent to

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