Geometric realization of the almost-extreme Khovanov homology of semiadequate links

TL;DR

This paper introduces a new geometric approach to understanding the almost-extreme Khovanov homology of semiadequate links, using partial presimplicial sets and their realizations.

Contribution

It defines partial presimplicial sets and demonstrates their use in constructing geometric realizations of Khovanov homology for semiadequate links, providing explicit formulas for their homotopy types.

Findings

Homotopy type involves wedge of spheres and a suspension of the projective plane.

Provides a concrete formula for the geometric realization.

Links semiadequate diagrams to topological constructions.

Abstract

We introduce the notion of partial presimplicial set and construct its geometric realization. We show that any semiadequate diagram yields a partial presimplicial set leading to a geometric realization of the almost-extreme Khovanov homology of the diagram. We give a concrete formula for the homotopy type of this geometric realization, involving wedge of spheres and a suspension of the projective plane.