The closure of a random braid is a hyperbolic link

Abstract:

Let $\mu$ be a probability distribution on the braid group $B_{n}$ with $n \geq 3$ strands. We observe that for a random walk $\omega _{n,k}$ of length $k$ on $B_{n}$, the probability that the closure $\widehat {\omega _{n,k}}$ is a hyperbolic link in $S^{3}$ converges to 1 as $k$ tends to infinity. Moreover, under a mild assumption on $\mu$, we prove the probability that the closure $\widehat {\omega _{n,k}}$ is a hyperbolic knot which has no non-trivial exceptional surgeries is larger than zero for $k$ large enough. The proofs combine several recent deep results.