Experiments on growth series of braid groups

Let $S$ be a finite generating set of a semigroup $M$. We denote by $S^{\ast}$ the set of all words on the alphabet $S$, which are called $S$-words. The empty word is denoted by $\varepsilon$. For every $S$-word $u$, we denote by $|u|$ its length and by $\overline{u}$ the element of $M$ it represents. We say that two $S$-words $u$ and $v$ are equivalent, denoted $u\equiv v$, is they represent the same element in $M$.

The $S$-length of an element $x\in M$ corresponds to the distance between $x$ and the identity in the Cayley graph of $M$ with respect to the finite generating set $S$.

If the language of geodesic $S$-words is regular then the series $\mathcal{G}(M,S)$ is rational.

If there exists a regular language composed of geodesic $S$-words in bijection with $M$ then the series $\mathcal{S}(M,S)$ is rational.

The first presentation of the braid group $B_{n}$ was given by E. Artin in [4] :

Artin’s presentation of $B_{n}$ implies that $\Sigma_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$ is a set of group generators of $B_{n}$. However the braid $\sigma_{\!1}^{\raise 0.8pt\hbox{$\scriptscriptstyle-$}1}$ cannot be represented by any $\Sigma_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$-word. For our purpose, it is fundamental to view a monoid (or a group) as a quotient of a finitely generated free monoid. As a monoid, the braid group $B_{n}$ is presented by generators $\Sigma_{n}$ and the relations of (1) plus relations

In [24], L. Sabalka constructed an explicit deterministic finite state automaton recognizing the language of geodesic $\Sigma_{3}$-words. He obtained the following rational value for the geodesic growth series of $B_{3}$ relatively to the Artin’s generators $\Sigma_{3}$ :

Moreover, using the finite state automaton recognizing the language of short-lex normal form of $B_{3}$ [17] he obtains :

The positive braid monoid $B_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$ is the submonoid of $B_{n}$ generated by $\Sigma_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$. Since every $\Sigma_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$-word is geodesic, the geodesic growth series $\mathcal{G}(B_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}},\Sigma_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}})$ is irrelevant. An explicit rational formula for the spherical growth series $\mathcal{S}(B_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}},\Sigma_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}})$ was obtained by A. Bronfman in [8] and later by M. Albenque in [1]. These results were extended to positive braid monoids of type B and D in [3] and for each Artin–Tits monoids of spherical type in [18].

In [7], J. Birman, K. H. Ko and S. J. Lee introduced a new generator family of $B_{n}$, called Birman-Ko-Lee’s or dual generators.

We write $[p,q]$ for the interval $\{p,\ldots,q\}$ of $\mathbb{N}$, and we say that $[p,q]$ is nested in $[r,s]$ if we have $r<p<q<s$.

Note that the definition of $a_{p,q}$ given here is not exactly that of [7] but it is coherent with previous papers of the author.

As for Artin’s generators, the braid group $B_{n}$ admits a monoid presentation with generators $\Sigma_{n}^{\scriptstyle*}$, relations (6) and (7) together with

Except in the case $n=2$, which is trivial, there are no results in the literature on the growth series of $B_{n}$ with respect to $\Sigma_{n}^{\scriptstyle*}$.

The Birman–Ko–Lee monoid $B_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*}$, also called dual braid monoid in [5] is the submonoid of $B_{n}$ generated by $\Sigma_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*}$. The term dual was used by D. Bessis since the Garside structure of $B_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$ and $B_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*}$ share symmetric combinatorial values. In [3], M. Albenque and P. Nadeau give a rational expression for the spherical growth series $\mathcal{S}(B_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*},\Sigma_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*})$; they also treat the case of dual braid monoids of type B.

The two monoids $B_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$ and $B_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*}$ equip the braid group $B_{n}$ with two Garside structures : the classical one [21] and the dual one [7, 5]. The reader can consult [14] and [13] for a general introduction to Garside theory. Here it is sufficient to know that each Garside structure provides simple elements which generate the corresponding Garside monoid. Let us denote by $C_{n}$ and $D_{n}$ the simple elements of the Garisde monoid $B_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$ and $B_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*}$ respectively.

In [11], P. Dehornoy starts the study of the spherical combinatorics of $B_{n}^{\raise 0.8pt\hbox{$\scriptscriptstyle+$}}$ relatively to $C_{n}$. In particular he formulates a divisibility conjecture which has been proven by F. Hivert, J.-C. Novelli and J.-Y. Thibon in [22]. A similar result was obtained for braid monoids of type B in [19]. The spherical combinatorics of $B_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*}$ relatively to $D_{n}$ was also considered by P. Biane and P. Dehornoy in [6]: they reduce the computation of $s(B_{n}^{\raise 0.6pt\hbox{$\scriptscriptstyle+$}*},D_{n};2)$ to that of free cumulants for a product of independent variables.

R. Charney establishes in [9] that the spherical growth series of Artin–Tits groups of spherical type with respect to their standard simple elements are rationals. In particular she obtains the rationality of $\mathcal{S}(B_{n},C_{n})$. This result was generalized for all Garside groups by P. Dehornoy in [10]. This implies in particular the rationality of $\mathcal{S}(B_{n},D_{n})$.

From an algorithmic point of view, a braid is naturally represented by a word. We extend this notion to any subset of $B_{n}(S_{n},\ell)$.

The previous example gives a representative set of $B_{n}(S_{n},\ell)$ for $\ell\leqslant 1$. We now tackle the construction of a representative set $W_{\ell}$ of $B_{n}(S_{n},\ell)$ for $\ell\geqslant 2$. Using an inductive argument we can assume we already have obtained a set $W_{\ell\mathchoice{\!-\!}{\!-\!}{\raise 0.7pt\hbox{$\scriptscriptstyle-$}\scriptstyle}{-}1}$ representing $B_{n}(S_{n},{\ell\mathchoice{\!-\!}{\!-\!}{\raise 0.7pt\hbox{$\scriptscriptstyle-$}\scriptstyle}{-}1})$ and then consider the set

A first step to obtain $W_{\ell}$ consists in removing all non-geodesic words from $W^{\prime}$. For this we have to test if a given word of $W^{\prime}$ is geodesic or not. A naive general solution consists in testing if a word $u\in W^{\prime}$ is equivalent to a $S_{n}$-word of length at most ${\ell\mathchoice{\!-\!}{\!-\!}{\raise 0.7pt\hbox{$\scriptscriptstyle-$}\scriptstyle}{-}1}$. However, as words of $W^{\prime}$ are obtained by appending a letter to a geodesic word, we can restrict the search space:

For all $\ell\in\mathbb{N}$ the number $g(B_{n},S_{n};\ell)$ can be obtained at no cost during the construction of a representative set of $B_{n}(S_{n},\ell)$.

We can now give a first algorithm returning a representative set $W_{\ell}$ of $B_{n}(S_{n},\ell)$ for $\ell\geqslant 2$. In order to determine $g(B_{n},S_{n};\ell)$ we also compute the value of $\omega_{S_{n}}$ for all words in $W_{\ell}$.

In order to construct by induction a representative set $W$, we must test if a given word $u$ is equivalent to a word occuring in $W$ :

In an algorithmic context a $S_{n}$-word is represented as an array of integers plus another integer $\omega$ which eventually correspond to $\omega_{S_{n}}(\overline{u})$. Whenever two variables u and v stand for the $S_{n}$-words $u$ and $v$ we use:

– $\texttt{u}\cdot\omega$ to design the integer $\omega$ associated to the word $u$;

– u v to design the product $uv$.

To be complete we must explain how to test if a $S_{n}$-word $u$ appears in a set of $S_{n}$-words. This can be achieved using a normal form (like the Garside’s normal form) but such a normal form doesn’t provide geodesic representatives. As, for our future research, we want to store braids using geodesic representatives, we prefer to use another method.

Originally defined in [16] from the geometric interpretation of the braid group $B_{n}$ as the mapping class group of the $n$-punctured disk of $\mathbb{R}^{2}$, the Dynnikov’s coordinates admit a purely algebraic definition from the action of $B_{n}$ on $\mathbb{Z}^{2n}$.

For $x\in\mathbb{Z}$, we denote by $x^{+}$ the non-negative integer $\max(x,0)$ and by $x^{-}$ the non-positive integer $\min(x,0)$. We first define an action of Artin’s generators on $\mathbb{Z}^{4}$.

We can now define an action of $\Sigma_{n}$-words on $\mathbb{Z}^{2n}$.

Naturally defined on braid words, Dynnikov’s coordinates is a braid invariant.

We now go back to the problem of testing if a given $S_{n}$-word appears in a set $W$ of $S_{n}$-words. A solution consists in representing the set $W$ in machine by an array. To test if the word $u$ appears in $W$ we can compute $\operatorname{Dyn}(u)$ and compare it to all the values of $\operatorname{Dyn}(v)$ whenever $v$ go through $W$. This method needs at most $1+\mathrm{card}\left(W\right)$ computations of Dynnikov’s coordinates. If words in $W$ are sorted by their Dynnikov’s coordinates we can test if $u$ appear in $W$ using at most $\log_{2}(\mathrm{card}\left(W\right))$ computations of Dynnikov’s coordinates. A more efficient solution is obtained using an unordered_set [26] based on a hash function. The insertion and lookup complexity is then constant in average on a RAM machine depending of the hash function.

As the objective of the current paper is to deepen our knowledge on combinatorics of $B_{4}$, we define a hash function for four strand braids. Assume $\beta$ is a braid of $B_{4}$ given by a $S_{4}$-word $u$. The hash of $\beta$ is

where $(a_{1},b_{1},\ldots,a_{4},b_{4})=\operatorname{Dyn}(u)$ and $\text{rem}(k,256)$ is the positive remainder of $k$ modulo $256$. By construction, $\text{hash}(\beta)$ is an integer lying in $[0,2^{64}-1]$ and so our hash function is very well suited for $64$ bits computers.

Here again we focus on the case $n=4$. The smallest addressable unit of memory on common computers is the byte which can have $256$ different values. As the set $\Sigma_{4}$ has $6$ elements we can store three $\Sigma_{4}$-letters using one byte ($6^{3}=216$). Hence a $\Sigma_{4}$-word of length $\ell$ requires $\lceil\frac{\ell}{3}\rceil$ bytes to be stored. Since there are $12$ elements in $\Sigma_{4}^{\scriptstyle*}$, a $\Sigma_{4}^{\scriptstyle*}$-word of length $\ell$ requires $\lceil\frac{\ell}{2}\rceil$ bytes to be stored.

Assume we want to determine a representative set of $B_{4}(\Sigma_{4},21)$. The memory needed by the algorithm RepSet is at least the space needed to store $\Sigma_{4}$-words of $W_{21}$. By Table 2 of Section 6 there are approximatively $60\cdot 10^{9}$ elements in this set. With the above storage method of a $\Sigma_{4}$-word, the algorithm needs $7\cdot 60\cdot 10^{9}$ bytes, i.e., $391$Go of memory to run, which is too much. To reduce the memory requirement we can split the sets $B_{n}(S_{n},\ell)$ in many subsets depending of the values of certain braid invariants.

In case we want to determine $g(B_{n},S_{n};\ell)$ we also store the value of $\omega_{S_{n}}(\overline{u})$ for all words in obtained representative sets.

For $n\geqslant 2$ we denote by $\mathfrak{S}_{n}$ the set of all bijections of $\{1,\ldots,n\}$ into itself. The transposition $(i\ {i\mathchoice{\!+\!}{\!+\!}{\raise 0.7pt\hbox{$\scriptscriptstyle+$}\scriptstyle}{+}1})$ of $\mathfrak{S}_{n}$ exchanging $i$ and ${i\mathchoice{\!+\!}{\!+\!}{\raise 0.7pt\hbox{$\scriptscriptstyle+$}\scriptstyle}{+}1}$ is denoted $s_{i}$.

If $\beta$ is a braid of $B_{n}$ then $\pi(\beta)$ is the permutation of $\mathfrak{S}_{n}$ such that the strand ending at position $i$ starts at position $\pi(\beta)(i)$.

As $\pi$ is a homomorphism, for all $\beta\in B_{n}$ and $x\in S_{n}$ we have $\pi(\beta\cdot x)=\pi(\beta)\circ\pi(x)$ and so the singleton $\{\pi\}$ is inductively stable.

Assume $\beta$ is a braid of $B_{n}$ and let $i$ and $j$ be two different integers of $[1,n]$. The linking number of the two strands $i$ and $j$ in $\beta$ is the algebraic number of crossings in $\beta$ involving the strands $i$ and $j$. A positive crossing ($\sigma_{\!k}$) counts for $+1$ whereas a negative one $(\sigma_{\!k}^{\raise 0.8pt\hbox{$\scriptscriptstyle-$}1})$ counts for $-1$ :

A priori, our definition of linking numbers depends of a diagram coding the braid and not on the braid itself. An immediate argument using relations (1) and (2) guarantees this is not the case. The reader can consult [12] page 29 for a more formal definition of linking number¹¹1In fact, the two definitions are slightly different but we have $\ell_{i,j}(\beta)=2\lambda_{i,j}(\beta)$. based of an integral definition and a geometric realization of $\beta$ in $\mathbb{R}^{3}$.

We now introduce the notion of template of a braid which will be used to parallelize the determination of a representative set of $B_{n}(S_{n},\ell)$.