The Heegaard structure of Dehn filled manifolds

TL;DR

This paper investigates how Dehn filling affects the Heegaard structure of 3-manifolds, analyzing changes like genus decrease, new surfaces, destabilization, and isotopy of surfaces, with classifications for torus knot exteriors.

Contribution

It provides a comprehensive survey of how Dehn filling influences Heegaard structures and classifies when specific phenomena occur, especially for torus knot exteriors.

Findings

Restrictions on genus decrease and new surface creation after filling.

Complete classification of phenomena in torus knot exteriors.

Conditions for destabilization and isotopy of Heegaard surfaces.

Abstract

We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilized Heegaard surface destabilizes, and (4) two or more non-isotopic Heegaard surfaces become isotopic. We survey general results that give quite satisfactory restrictions to phenomena (1) and (2) and, in a parallel thread, give a complete classification of when all four phenomena occur when filling most torus knot exteriors. This latter thread yields sufficient (and perhaps necessary) conditions for the occurrence of phenomena (3) and (4).