Bott-Cattaneo-Rossi invariants for long knots in asymptotic homology $\mathbb{R}^{3}$

Let ${M}$ be a smooth oriented compact connected $3$-manifold with the rational homology of $\mathbb{S}^{3}$. Fix a point $\infty$ in ${M}$ and a closed ball $B_{\infty}(M)$ around $\infty$, and set ${{{M}^{\circ}}}={M}\setminus\{\infty\}$. Identify the punctured ball $B_{\infty}^{\circ}(M)=B_{\infty}(M)\setminus\{\infty\}$ of ${{{M}^{\circ}}}$ with the complement ${B_{\infty}()^{\circ}}$ of the open unit ball in $\mathbb{R}^{3}$, and let $B(M)$ denote the closure of ${{{M}^{\circ}}}\setminus{B_{\infty}()^{\circ}}$. The manifold ${{{M}^{\circ}}}$ together with the decomposition ${{{M}^{\circ}}}=B(M)\cup{B_{\infty}()^{\circ}}$ is called an asymptotic rational homology $\mathbb{R}^{3}$.

A parallelization of an asymptotic rational homology $\mathbb{R}^{3}$ is a bundle isomorphism $\tau\colon{{{M}^{\circ}}}\times\mathbb{R}^{3}\rightarrow T{{{M}^{\circ}}}$ that coincides with the canonical trivialization of $T\mathbb{R}^{3}$ on ${B_{\infty}()^{\circ}}\times\mathbb{R}^{3}$. Such a parallelization always exists : see for example [Les20, Proposition 5.5].

A long knot in an asymptotic rational homology $\mathbb{R}^{3}$ is a smooth embedding $\psi\colon\mathbb{R}\hookrightarrow{{{M}^{\circ}}}$ such that, for any $x\in\mathbb{R}$,

• •

  if $x\in[-1,1]$, $\psi(x)\in B(M)$,

• •

  if $x\not\in[-1,1]$, $\psi(x)=(0,0,x)\in{B_{\infty}()^{\circ}}\subset\mathbb{R}^{3}$.

In the following, we let $({{{M}^{\circ}}},\tau)$ be a fixed parallelized rational asymptotic homology $\mathbb{R}^{3}$.

We recall the definition of BCR diagrams, as introduced in [Let19, Section 2.2]. In all the following, if $k$ is a positive integer, $\underline{k}$ denotes the set $\{1,\ldots,k\}$.

In the following, internal edges are depicted by solid arrows, external edges by dashed arrows, internal vertices by black dots, and external vertices by white dots, as in Figure 1, where the five behaviors of Definition 2.1 appear.

Definition 2.1 implies that any BCR diagram consists of one cycle with some legs attached to it, where legs are external edges that come from a (necessarily internal) univalent vertex, and where the graph is a cyclic sequence of pieces as in Figure 2 with as many pieces of the first type than of the second type. In particular, a BCR diagram has an even number of vertices, and this number is also the number of its edges.

The degree of a BCR diagram is the integer $\deg(\Gamma)=\frac{1}{2}{\mathrm{Card}}(V(\Gamma))$. A numbering⁶⁶6In this article, numberings are valued in $\underline{3k}$ rather than in $\underline{2k}$ as in [Let19] or [Let20], and only the external edges are numbered. of a degree $k$ BCR diagram is an injection $\sigma\colon E_{e}(\Gamma)\hookrightarrow\underline{3k}$.

In this section, we recall the definition of unitrivalent diagrams, widely used in the theory of Vassiliev invariants.

In the following, Jacobi diagrams will be depicted by a planar immersion of $\Gamma\cup\mathbb{R}$, where $\mathbb{R}$ is a plain vertical line, on which the univalent vertices of $\Gamma$ lie, and the edges of $\Gamma$ are dashed lines. The order on the univalent vertices is given by the vertical direction on the plain vertical line $\mathbb{R}$ from bottom to top. When dealing with vertex-oriented Jacobi diagrams, the vertex-orientation will be given by the counterclockwise order in the plane. An example of Jacobi diagram is given in Figure 3.

The product of Jacobi diagrams is compatible with the relations of Definition 2.3. Bar-Natan proved in [BN95a, Theorem 7] that the induced graded algebra structure on $\prod\limits_{k\in\mathbb{N}}\mathcal{A}_{k}$ is commutative.⁷⁷7Bar-Natan’s theorem proves that $\prod\limits_{k\in\mathbb{N}}\mathcal{A}_{k}$ is a commutative and cocommutative Hopf algebra, for a given coproduct, but we do not use the coproduct in this article. This allows us to define the associated exponential map $\exp\colon\prod\limits_{k\in\mathbb{N}}\mathcal{A}_{k}\rightarrow\prod\limits_{k\in\mathbb{N}}\mathcal{A}_{k}$.

Let us first recall the definition of the Conway weight system of [BNG96, Section 3.1].

In [BNG96], Bar-Natan and Garoufalidis proved that this definition determines a linear form $w_{C}\colon\bigoplus\limits_{k\in\mathbb{N}}{\check{\mathcal{A}_{k}}}\rightarrow\mathbb{R}$. Set $w_{C}(\Gamma)=0$ for diagrams with at least one component without univalent vertices. The form $w_{C}$ naturally extends to $w_{C}\colon\bigoplus\limits_{k\in\mathbb{N}}{\mathcal{A}_{k}}\rightarrow\mathbb{R}$ by sending diagrams with a non-empty trivalent⁸⁸8A trivalent graph is a graph with only trivalent vertices. connected component to zero. Chmutov [Chm98, p. 9] proved that $w_{C}$ is determined by the following properties.

The following result directly follows from [Les20, Corollary 6.36].

Define the logarithmic Conway weight system $w^{\prime}_{C}$ as $w^{\prime}_{C}=w_{C}\circ p^{c}$. It is characterized as follows.

Here, we review the definition of $C_{2}({{{M}^{\circ}}})$ in [Les15, Section 2.2]. Let $C_{2}({{{M}^{\circ}}})$ denote the space obtained from $M^{2}$ after the differential blow-up ⁹⁹9For example, see [Les15, Section 2.2] for more details on these blow-ups. of $\{(\infty,\infty)\}$ and of the closures of ${{{M}^{\circ}}}\times\{\infty\}$, $\{\infty\}\times{{{M}^{\circ}}}$ and $\Delta=\{(x,x)\mid x\in{{{M}^{\circ}}}\}$ in the obtained manifold. The manifold with boundary and corners $C_{2}({{{M}^{\circ}}})$ is a compactification of $C_{2}^{0}({{{M}^{\circ}}})=({{{M}^{\circ}}})^{2}\setminus\Delta$. Let $G_{\tau}\colon\partial C_{2}({{{M}^{\circ}}})\rightarrow\mathbb{S}^{2}$ denote the Gauss map as defined in¹⁰¹⁰10It is called $p_{\tau}$ in these sources. [Les15, Proposition 2.3] or [Les20, Proposition 3.7]. The map $G_{\tau}$ is an analogue of the map $G\colon C_{2}(\mathbb{R}^{3})\rightarrow\mathbb{S}^{2}$ that extends $\big{(}(x,y)\in C_{2}^{0}(\mathbb{R}^{3})\mapsto\frac{y-x}{||y-x||}\in\mathbb{S}^{2}\big{)}$, but $G_{\tau}$ is only defined on the boundary of the two-point configuration space. It depends on the parallelization $\tau$ of Section 2.1.

A form ${\omega}$ on $C_{2}({{{M}^{\circ}}})$ is antisymmetric if $T^{*}({\omega})=-{\omega}$, where $T\colon C_{2}({{{M}^{\circ}}})\rightarrow C_{2}({{{M}^{\circ}}})$ is the smooth extension of $(x,y)\in C_{2}^{0}({{{M}^{\circ}}})\mapsto(y,x)\in C_{2}({{{M}^{\circ}}})$. Let us recall the definition of the external propagators of [Let19, Section 2.5]. ¹¹¹¹11Here, we require antisymmetric propagators in order to get a simpler formula in Theorem 2.13.

In $\mathbb{R}^{3}$ with its canonical parallelization, the pull-back of the $SO(3)$-invariant form on $\mathbb{S}^{2}$ with total volume $1$ under the Gauss map $G$ is a propagator.

For any degree $k$ numbered BCR diagram $(\Gamma,\sigma)$ and any $k$-family $F=({\omega}_{i})_{i\in\underline{3k}}$ of propagators of $({{{M}^{\circ}}},\psi,\tau)$, define a form $\omega^{F}(\Gamma,\sigma,\psi)$ on $C^{0}_{\Gamma,BCR}(\psi)$ as

$$\omega^{F}(\Gamma,\sigma,\psi)=\frac{(-1)^{N_{i}^{-}(\Gamma,\cdot)}}{2^{{\mathrm{Card}}(E_{i}(\Gamma))}}\bigwedge\limits_{e\in E_{e}(\Gamma)}{p_{e}}^{*}({\omega}_{\sigma(e)}),$$

where for any $c\in C_{\Gamma}(\psi)$, $N_{i}^{-}(\Gamma,c)$ is the number of internal edges from a vertex $v$ to a vertex $w$ such that $\varepsilon_{v,w}(c)<0$. Note that the form $\omega^{F}(\Gamma,\sigma,\psi)$ is the analogue of the form of the same name defined in [Let19, Section 2.6], when all the internal propagators of this article are equal to the $0$-form $\alpha$ on $C_{2}(\mathbb{R})$ that extends $\left((x,y)\in C_{2}^{0}(\mathbb{R})\mapsto\frac{\mathrm{sgn}(y-x)}{2}\in\mathbb{R}\right)$.

For any degree $k$ numbered edge-oriented Jacobi diagram $(\Gamma,j)$ and any $k$-family $F=({\omega}_{i})_{i\in\underline{3k}}$ of propagators of $({{{M}^{\circ}}},\tau)$, define a form $\omega^{F}(\Gamma,j,\psi)$ as

$$\omega^{F}(\Gamma,j,\psi)=\bigwedge\limits_{e\in E(\Gamma)}{p_{e}}^{*}({\omega}_{j(e)}).$$

Now, for any numbered BCR diagram, set

$$I^{F}(\Gamma,\sigma,\psi)=\int_{C_{\Gamma,BCR}^{0}(\psi)}\omega^{F}(\Gamma,\sigma,\psi),$$

and for any numbered bioriented Jacobi diagram, set

$$I^{F}(\Gamma,j,\psi)=\int_{C_{\Gamma,J}^{0}(\psi)}\omega^{F}(\Gamma,j,\psi).$$

These two integrals converge because of the existence of compactifications of the configuration spaces, to which the above forms extend.¹²¹²12Such compactifications were first introduced by Axelrod and Singer in [AS94, Section 5] for Jacobi diagrams and by Rossi in his thesis [Ros02, Section 2.5] for BCR diagrams. See also [Les20, Section 8] for an extensive description of these compactifications for Jacobi diagrams.

Let $\widetilde{\mathcal{D}_{k}}$ denote the set of degree $k$ (unoriented) numbered Jacobi diagrams. Let $(\Gamma,j)\in\widetilde{\mathcal{D}_{k}}$, and let $F=(\omega_{i})_{i\in\underline{3k}}$ be a family of propagators of $({{{M}^{\circ}}},\tau)$. Orient both the edges and the trivalent vertices of $\Gamma$ arbitrarily. Since the propagators are antisymmetric, the orientation of $C_{\Gamma,J}^{0}(\psi)$ and the sign of $\omega^{F}(\Gamma,j,\psi)$ both change when changing the orientation of one edge of $\Gamma$, and $w^{\prime}_{C}(\Gamma)$ remains unchanged. When the orientation of one trivalent vertex changes, the sign of $w^{\prime}_{C}([\Gamma])$ and the orientation of $C_{\Gamma,J}^{0}(\psi)$ both change and $\omega^{F}(\Gamma,j,\psi)$ remains unchanged. Therefore, $I^{F}(\Gamma,j,\psi)w^{\prime}_{C}([\Gamma])$ depends neither of the vertex-orientation, nor of the edge-orientation, and is thus well-defined for $(\Gamma,j)\in\widetilde{\mathcal{D}_{k}}$.

Let $j_{\emptyset}$ denote the only numbering of the empty diagram, let $F_{\emptyset}$ denote the empty family of propagators, and set $I^{F_{\emptyset}}(\emptyset,j_{\emptyset},\psi)=1$.

The following theorem is the main result of this article. It is proved in Section 3.

Note that the coefficients $\frac{(3k-{\mathrm{Card}}(E_{e}(\Gamma)))!}{(3k)!}$ in the definition of $Z^{F}_{BCR,k}$ replace the $\frac{1}{(2k)!}$ in the definition of [Let19, Theorem 2.10] since we allowed numberings to take value in $\underline{3k}$ and since the internal edges are not numbered anymore.

The above theorem and Theorem 2.13 imply the following corollary.

Since the propagators defined in [Let20, Section 4] are dual to propagators in the sense of Definition 2.11, [Let20, Theorem 2.31] implies the following theorem.

The above theorem and Theorems 2.13 and 2.14 imply the following.

In this section, we prove that Corollary 2.17 can be stated equivalently in terms of the Kontsevich integral or of the perturbative expansion of the Chern-Simons theory, for long knots of $\mathbb{R}^{3}$.