Bridge numbers of knots in the page of an open book

TL;DR

This paper demonstrates that in any closed, orientable 3-manifold, there exist knots with arbitrarily large genus g bridge numbers within a page of an open book decomposition, using monodromy actions on the arc and curve complex.

Contribution

It establishes the existence of knots with arbitrarily large genus g bridge numbers in open book pages, extending understanding of knot complexity in 3-manifolds.

Findings

Existence of knots with large genus g bridge number in any 3-manifold.

Construction of such knots within open book pages.

Identification of Berge knots with arbitrarily large genus one bridge number.

Abstract

Given any closed, connected, orientable $3$--manifold and integers $g\geq g(M), D > 0$, we show the existence of knots in $M$ whose genus $g$ bridge number is greater than $D$. These knots lie in a page of an open book decomposition of $M$, and the proof proceeds by examining the action of the map induced by the monodromy on the arc and curve complex of a page. A corollary is that there are Berge knots of arbitrarily large genus one bridge number.