Normal surfaces and colored Khovanov homology

TL;DR

This paper demonstrates that colored Khovanov homology can detect essential surfaces in knot complements, linking it to the slope conjectures and using ideal triangulations to identify chain complex generators.

Contribution

It introduces a method to identify generators of colored Khovanov homology with normal surfaces, establishing a new connection between homology and geometric topology.

Findings

Colored Khovanov homology detects classes of essential surfaces.

Identification of chain complex generators with normal surfaces.

Extension of slope conjectures to colored Khovanov homology.

Abstract

We show that colored Khovanov homology detects classes of essential surfaces as a direct analogue of the slope conjectures for the colored Jones polynomial. We do this by identifying certain generators of the colored Khovanov chain complex with normal surfaces in the complement of the knot using an ideal triangulation from a diagram.