Computing Khovanov homology of tangles

TL;DR

This paper introduces a new arc reduction method for computing Khovanov homology of tangles, providing explicit calculations and Poincaré polynomials for simple and low-crossing tangles, advancing computational knot theory.

Contribution

The paper presents a novel arc reduction approach for calculating Khovanov homology of tangles and derives Poincaré polynomials for tangles with up to three crossings.

Findings

Derived Poincaré polynomials for simple tangles.

Computed Khovanov homology for tangles with up to three crossings.

Introduced an arc reduction method for tangle homology computation.

Abstract

The computation of Khovanov homology for tangles has significant potential applications, yet explicit computational studies remain limited. In this work, we present a method for computing the Khovanov homology of tangles via an arc reduction approach, and we derive the Poincar\'e polynomial for simple tangles. Furthermore, we compute the Poincar\'e polynomials of tangles with at most three crossings.