A quantitative Birman-Menasco finiteness theorem and its application to crossing number

In [7] Birman-Menasco proved (in)finiteness theorem on closed braid representatives of knots and links; a knot or link $K$ has only finitely many closed braid representatives, up to conjugacy and exchange move. Here an exchange move is an operation that converts an $n$-braid of the form $\alpha\sigma_{n-1}\beta\sigma_{n-1}^{-1}$ ($\alpha,\beta\in B_{n-2}$) to a braid $\alpha\sigma_{n-1}^{-1}\beta\sigma_{n-1}$ and its converse (see Figure 1). As a byproduct they proved another remarkable finiteness result;

In this paper, we give a quantitative version of Theorem 1, an estimate of the crossing number by braid index and genus. For a link $L$ in $S^{3}$, let $b(L),c(L)$ and $\chi(L)$ be the braid index, the minimum crossing number, and the maximum euler characteristic of Seifert surface, respectively.

The case $b(L)=2$ is easy and the case $b(L)=3$ is proven in [13] by using a special property of closed 3-braid that a closed 3-braid always bounds a minimum genus Bennequin surface, a minimum genus Seifert surface whose braid foliation has only aa-singular points [1, 5].

In the following, we restrict our attention to knots, and use the knot genus instead of the maximum euler characteristic.

Theorem 2 implies that $2g(K)-1+b(K)$ is a linear approximation of the crossing number. This has several applications to the crossing numbers since the genus is one of the most familiar and well-studied invariant whereas the minimum crossing number is one of the most difficult invariant whose basic properties are still unknown.

It is a famous conjecture that $c(K\#K^{\prime})=c(K)+c(K^{\prime})$ [14, Problem 1.65]. Theorem 2 immediately gives an estimate of the crossing number of composite knots.

In [15] Lackenby showed that $c(K\#K^{\prime})\geq\frac{1}{152}(c(K)+c(K^{\prime}))$. Although the estimate given in Corollary 1 gets worse as $b(K)$ increases, when $b(K)$ is small it gives a better estimate.

Our proof gives an estimate of regularity introduced in [18] which plays an important role to relate two different problem; a genericity of hyperbolic knots/links and the additivity of crossing numbers.

Similarly, we have an estimate of the crossing number of satellite knots and an estimate of the asymptotic crossing number. The asymptotic crossing number $ac(K)$ of $K$ is defined by¹¹1The definition of the asymptotic crossing number given here is a version appeared in Kirby’s problem list [14]. In [8] Freedman-He called this the quadratic asymptotic crossing number and use different definition for the assymptotic crossing number, which we denote by $ac_{FH}(K)$. Since Freedman-He proved $ac(K)\geq ac_{FH}(K)\geq 2g(K)-1$, the inequality in Corollary 3 is valid for $ac_{FH}(K)$.

where the infimum is taken over all satellites $K_{w}$ of $K$ with winding number $w\geq 0$. It is conjectured $ac(K)=c(K)$ [14, Problem 1.68].

Although Corollary 3 is meaningful only if $b(K)$ is small compared with $c(K)$, it gives a linear lower bound of $ac(K)$ in terms of $c(K)$.

A famous satellite crossing number conjecture asserts $c(K)\geq c(K_{0})$ [14, Problem 1.67]. Corollary 3 says that the satellite crossing number conjecture is true when $w$ and $c(K_{0})$ are large enough compared with $b(K_{0})$.

A weaker and more tractable version of the satellite crossing number conjecture, $c(K_{p,q})\geq c(K)$ for the $(p,q)$-cable $K_{p,q}$ of $K$ which we call the cabling crossing number conjecture, is still open. For a braided $p$-cable, a satellite knot whose pattern $C$ is a closed $p$-braid in the solid torus, we have another estimate that gives more supporting evidences for the cabling crossing number conjecture and the satellite crossing number conjecture which works even if $b(K)$ is large;

In particular, since $f(3)=\frac{5}{3}<2$, the cabling crossing number conjecture is true for closed 3-braids:

It is interesting to ask whether Corollary 4 can be extended to general satellite knot. We used the braided $p$-cable assumption is to guarantee $b(K_{p})=pb(K)$. It looks to be reasonable to expect $b(K_{w})\geq wb(K)$ for satellite $K_{w}$ of $K$ with winding number $w$ so the same conclusion as Corollary 4 holds for general satellite knot $K_{w}$ with winding number $w$.

In [16] Lackenby showed $c(K)\geq\frac{1}{10^{13}}c(K_{0})$ if $K$ is a satellite of $K_{0}$. Since this $\frac{1}{10^{13}}$ estimate uses an estimate for $(2,t)$-cable of $K_{2,t}$ of $K_{0}$ and Corollary 4 gives an improvement of Lackenby’s estimate $\frac{1}{119024}(c(K_{0})-|t|)\leq c(K_{2,t})$ [16, Theorem 5.1], one can improve Lackenby’s $\frac{1}{10^{13}}$ bound when $b(K_{0})$ is not so large. Indeed, combining Lackenby’s another result and our result we give a dramatic improvement of the estimate of the crossing number of satellite knots when $b(K_{0})$ is not large.

Finally we give another application. The braid index problem is a problem to algorithmically compute the braid index of a link $L$ from a given diagram $D$ of $L$. Strictly speaking, to make it as a decision problem, by the braid index problem we mean the problem to determine, for a given diagram $D$ of $L$ and a natural number $n$, determine whether $b(L)\leq n$ or not. Since $K$ is the unknot if and only if $b(K)=1$, this problem includes the unknotting problem, a problem to recognize the unknot.

It is desirable to give more direct algorithm of the braid index problem that avoids to use knot recognition, in a spirit of the Birman-Hirsch unknotting algorithm [3]. Also, it is an interesting problem to determine the computational complexity of the braid index problem.

The author has been partially supported by JSPS KAKENHI Grant Number 19K03490,16H02145. He would like to thank Joan Birman for stimulating comments for earlier version of the paper, and would like to thank to Sorsen Rebecca and Keiko Kawamuro pointing out an error in the earlier version of the paper.