Random walks and the symplectic representation of the braid group

TL;DR

This paper studies the properties of random walks on braid groups through their symplectic representations, revealing transience of certain polynomial conditions, probabilities of p-colorability, and characteristics of quasipositive links and Lissajous knots.

Contribution

It introduces new results on the transience of polynomial conditions in symplectic representations of braid groups and analyzes probabilistic properties of braids related to knot colorability and signatures.

Findings

Polynomial conditions define transient sets for random walks on braid groups.

Probability of a braid closing into a p-colorable loop exceeds 1/p.

Quasipositive 3-braids have zero signature for all integer powers.

Abstract

We consider the symplectic representation $\rho_n$ of a braid group $B(n)$ in $Sp(2l,\mathbb{Z})$, for $l=\Big[\dfrac{n-1}{2}\Big]$. If $P$ is a polynomial on the $4l^2$ coefficients of the matrices in $Sp(2l,\mathbb{Z})$, we show that the set $\{\beta\in B(n): P(\rho_n(\beta))=0\}$ is transient for non degenerate random walks on $B(n)$. We derive that the $n$-braids $\beta$ which close into a loop $\hat{\beta}$ with $0<|det({\hat{\beta}})|\leq C$ for some constant $C$ form a transient set. And given a prime number $p$, we show that the probability for a given braid to close in a $p$-colorable loop is greater than $\dfrac{1}{p}$. We also derive that for a random $3$-braid, the quasipositive links $(\beta\sigma_i\beta^{-1}\sigma_j)^p$ have zero signature for every integer $p$ and $1\leq i,j\leq 2$. \\ As an example of such braids, we investigate the signature of the Lissajous toric knots $3$-braids.