A non-degenerate exchange move always produces infinitely many non-conjugate braids

TL;DR

This paper proves that if a link has a closed braid with a non-degenerate exchange move, it must have infinitely many non-conjugate braid representations, revealing complex structure in braid conjugacy classes.

Contribution

It establishes that non-degenerate exchange moves on closed braids imply infinitely many non-conjugate braid representations of the link.

Findings

Non-degenerate exchange moves lead to infinitely many non-conjugate braids.

Links with such moves have complex conjugacy class structures.

The result applies to closed n-braids representing links.

Abstract

We show that if a link $L$ has a closed $n$-braid representative admitting non-degenerate exchange move, an exchange move that does not obviously preserve the conjugacy class, $L$ has infinitely many non-conjugate closed $n$-braid representatives.

Equations

$${\mathrm{ex}}^k(\beta {)}^N{|}_S=T_{c_1}^k{T}_{c_2}^{-k}\cdots T_{c_{2N-1}}^k{T}_{c_{2N}}^{-k}{\beta }^N{|}_S\in MCG(S)$$

$$\mathrm{ent}({\mathrm{ex}}^k(\beta ))$$

Full text

Let $B_{n}$ be the braid group with standard generators $\sigma_{1},\ldots,\sigma_{n-1}$. We denote the closure of a braid $\beta\in B_{n}$ by $\widehat{\beta}\subset S^{3}$. For an oriented link $L\subset S^{3}$ let $\mathbf{Br}_{n}(L)=\{\beta\in B_{n}\>|\>\widehat{\beta}=L\}$ be the set of $n$-braids whose closures are $L$.

The set $\mathbf{Br}_{n}(L)$ may contain infinitely many mutually non-conjugate braids. However, Birman and Menasco proved a remarkable (non)finiteness theorem [BM]: $\mathbf{Br}_{n}(L)$ modulo exchange move $\alpha\sigma_{n-1}\beta\sigma_{n-1}^{-1}\leftrightarrow\alpha\sigma_{n-1}^{-1}\beta\sigma_{n-1}$ ($\alpha,\beta\in B_{n-1}$) has only finitely many conjugacy classes. In particular, when $\mathbf{Br}_{n}(L)$ does not contain a braid admitting exchange move, $\mathbf{Br}_{n}(L)$ contains only finitely many conjugacy classes.

Then we ask the converse111We need to be bit careful to formulate the problem since some exchange moves are ‘trivial’ in the sense that they obviously yield conjugate braids.: Does $\mathbf{Br}_{n}(L)$ contain infinitely many mutually non-conjugate braids if $\mathbf{Br}_{n}(L)$ contains a braid admitting exchange move ?

This question was studied in [SS, Sh, St1, St2] where it was shown that under some additional and technical assumptions, iterations of exchange move indeed produce infinitely many non-conjugate braids.

In this note we give a simpler and shorter proof of infiniteness under the weakest assumption. We use a formulation of iterations of exchange moves following [SS]:

For $k\in\mathbb{Z}$ and an $n$-braid $\beta=AB$ admitting an exchange move, the $k$-iterated exchange move of $\beta=AB$ is the braid $\mathrm{ex}^{k}(\beta)=A\tau^{k}B\tau^{-k}$ where $\tau=(\sigma_{2}\cdots\sigma_{n-2})^{n-2}\in B_{n}$. We say that an (iterated) exchange move is degenerate if $A\tau=\tau A$ or $B\tau=\tau B$. Otherwise, an (iterated) exchange move is non-degenerate.

A $k$-iterated exchange move is attained by exchange move $|k|$ times so the closures of $\beta$ and $\mathrm{ex}^{k}(\beta)$ represent the same link. A degenerate exchange move is an exchange move that obviously preserves the conjugacy classes. Our main theorem shows that, except this trivial cases, iterated exchange moves always produce infinitely many mutually non-conjugate braids.

We identify the braid group $B_{n}$ with the mapping class group $MCG(D_{n})$ of the $n$-punctured disk $D_{n}$. Let $\mathrm{ent}(\beta)$ be the topological entropy of $\beta$, the infimum of the topological entropy of homeomorphism that represents $\beta$.

Let $S$ be a closed orientable surface minus finitely many disks and puncture points in its interior. A simple closed curve $c$ in $S$ is essential if $c$ is not boundary parallel nor surrounding single puncture. We denote by $T_{c}$ the Dehn twist along $c$. A family of essential simple closed curves $\{c_{1},\ldots,c_{N}\}$ fills $S$ if $\min i(c,c_{i})\neq 0$ for any essential simple closed curve $c$, where $i(c,c^{\prime})$ denotes the geometric intersection number. Our proof is based on the following theorem of Fathi [Fa, Theorem 7.9].

1. (i)

The set of curves $\{c_{1},\ldots,c_{N}\}$ fills $S$. 2. (ii)

$i(c_{i},c_{i+1})\neq 0$* for $i=1,\ldots,N-1$ and $i(c_{N},c_{1})\neq 0$.*

Then for given $R>0$, there is $k=k(R)>0$ such that $T_{c_{1}}^{n_{1}}T_{c_{2}}^{n_{2}}\cdots T_{c_{N}}^{n_{N}}f$ is pseudo-Anosov whose dilation is $>R$ whenever $|n_{i}|>k$ for all $i$.

For $i>0$ let $c_{2i}=\beta^{i-1}(A(c))=(AB)^{i-1}A(c),c_{2i-1}=\beta^{i-1}(c)=(AB)^{i}(c)$. Then for $N>0$, $\mathrm{ex}^{k}(\beta)^{N}=T_{c_{1}}^{k}T_{c_{2}}^{-k}\cdots T_{c_{2N-1}}^{k}T_{c_{2N}}^{-k}\beta^{N}$.

We equip a complete hyperbolic metric (of finite area) on $D_{n}$ and take $c_{i}$ as a closed geodesic. Let $S$ be the minimum complete geodesic subsurface of $D_{n}$ that contains $\{c_{1},c_{2},c_{3},c_{4},\ldots\}=\{c,A(c),\beta(c),\beta(A(c)),\ldots\}$. Here the minimum means the minimum with respect to inclusions. Then for sufficiently large $M$ the set of curves $\{c_{1},c_{2},\ldots,c_{M}\}$ fills $S$. By non-degeneracy assumption, $S$ contains $\partial D_{n}$ as its boundary.

By non-degeneracy assumption $i(c_{i},c_{i+1})\neq 0$ for every $i>0$. Thus by Claim 2 there is $N>0$ such that such

[TABLE]

satisfies the assumptions of Theorem 3. Hence for any given $R>0$, whenever $|k|$ is sufficiently large, $\mathrm{ex}^{k}(\beta)^{N}|_{S}$ is pseudo-Anosov whose dilatation $\lambda(\mathrm{ex}^{k}(\beta)^{N}|_{S})$ is $>R$. Since

[TABLE]

the set $\{\mathrm{ent}(\mathrm{ex}^{k}(\beta))\>|\>k\in\mathbb{Z}\}$ is unbounded. ∎

Our proof shows that as $k$ increase, a non-degenerate $k$-iterated exchange move increases the entropy, the complexity of dynamics, as long as $k$ is sufficiently large. Since the core of Birman-Menasco’s proof of (non)finiteness theorem is to reduce the complexity (the number of singular points) of braid foliation of Seifert surface, it is natural to expect relations between the entropy and braid foliation.

Acknowledgement

The author is partially supported by JSPS KAKENHI Grant Numbers19K03490, 16H02145. This research was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program.