Independence complexes of circle graphs

TL;DR

This paper investigates the homotopy types of independence complexes of circle graphs, especially bipartite ones, and connects these combinatorial structures to topological properties of links via Khovanov homology, including explicit computations for pretzel knots.

Contribution

It provides new insights into the homotopy types of independence complexes of circle graphs and links these to topological invariants of knots, with explicit computations for pretzel knots.

Findings

Homotopy types of independence complexes are characterized for circle graphs.

Bipartite circle graphs' independence complexes are analyzed.

Explicit computation of Khovanov homology for a 4-strand pretzel knot.

Abstract

Independence complexes of circle graphs are purely combinatorial objects. However, when constructed from some diagram of a link $L$, they reveal topological properties of $L$, more specifically, of its Khovanov homology. We analyze the homotopy type of independence complexes of circle graphs, with a focus on those arising when the graph is bipartite. Moreover, we compute (real) extreme Khovanov homology of a $4$-strand pretzel knot using chord diagrams and independence complexes.