Surgery obstructions from Khovanov homology

TL;DR

This paper demonstrates how Khovanov homology can be used to obstruct certain types of Dehn surgeries on 3-manifolds with torus boundary, providing new tools for understanding 3-manifold topology.

Contribution

It introduces novel obstructions derived from Khovanov homology for lens space and finite fundamental group surgeries on knots, especially strongly invertible knots in S^3.

Findings

Khovanov homology provides effective obstructions to certain Dehn fillings.

Homological width in Khovanov homology is key to these obstructions.

Explicit calculations for Montesinos links support the theoretical results.

Abstract

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.