High distance knots in closed 3-manifolds

TL;DR

This paper demonstrates that after a single stabilization, some core of a Heegaard splitting in a closed 3-manifold can have arbitrarily high distance, extending previous results and exploring properties of handlebody sets and the coarse mapping class group.

Contribution

It generalizes a known result about knots in S^3 to all closed 3-manifolds and introduces the concept of the coarse mapping class group of a Heegaard splitting.

Findings

Existence of cores with arbitrarily high distance after stabilization

Handlebody sets are either coarsely distinct or identical in the curve complex

Coarse mapping class group is isomorphic to the classical mapping class group for genus ≥ 2

Abstract

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus greater than or equal to 2, then the coarse mapping class group of (S,V,W) is isomorphic to the mapping class group of (S, V,W).