Stabilizing Heegaard Splittings of High-Distance Knots

TL;DR

This paper investigates the structure and stabilization properties of minimal genus Heegaard splittings of knot complements in $S^3$, establishing bounds on their equivalence after stabilization based on the knot's bridge distance.

Contribution

It provides bounds on the number and stabilization behavior of minimal genus Heegaard splittings for knots with high bridge distance, introducing a classification into two families.

Findings

At most ${2nrace n}$ minimal genus Heegaard splittings for knots with bridge distance > 2n.

Splittings within the same family become equivalent after at most one stabilization.

Splittings from different families require up to $n-1$ stabilizations to become equivalent when bridge distance ≥ 4n.

Abstract

Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminus\eta(K)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to these Heegaard surfaces.