On tunnel number one knots that are not (1,n)

TL;DR

This paper demonstrates that for any natural number, there exist tunnel number one knots in the 3-sphere that cannot be represented as (1,n) knots, by establishing a lower bound on bridge number related to Heegaard splitting distance.

Contribution

It introduces a new lower bound on the bridge number of t-bridge knots based on Heegaard splitting distance, showing the existence of tunnel number one knots not equivalent to (1,n) knots.

Findings

Existence of tunnel number one knots not representable as (1,n)

Bridge number bounded below by a function of Heegaard splitting distance

For any natural number n, such knots exist

Abstract

We show that the bridge number of a $t$ bridge knot in $S^3$ with respect to an unknotted genus $t$ surface is bounded below by a function of the distance of the Heegaard splitting induced by the $t$ bridges. It follows that for any natural number $n$, there is a tunnel number one knot in $S^3$ that is not $(1,n)$.