Khovanov homology of tangles: algorithm and computation

TL;DR

This paper introduces a new topological quantum field theory approach to compute Khovanov homology for tangles, providing detailed algorithms and practical implementation guidance to expand applications in various scientific fields.

Contribution

It presents a comprehensive method and algorithm for computing Khovanov homology of tangles, which was previously underexplored, using a TQFT framework.

Findings

Developed a TQFT-based construction for tangle Khovanov homology

Provided detailed algorithms and coding guidance for computation

Facilitated broader application potential in science and mathematics

Abstract

Knot, link, and tangle theory is crucial in both mathematical theory and practical application, including quantum physics, molecular biology, and structural chemistry. Unlike knots and links, tangles impose more relaxed constraints, allowing the presence of arcs, which makes them particularly valuable for broader applications. Although Khovanov homology for knots and links has been extensively studied, its computation for tangles remains largely unexplored. In our recent work, we provide a topological quantum field theory (TQFT) construction for the Khovanov homology of tangles, offering a more concrete method for its computation. The primary contribution of this work is a comprehensive approach to the computation of the Khovanov homology of tangles, offering both a detailed computation procedure and a practical guide for implementing algorithms through codes to facilitate the calculation. This contribution paves the way for further studies and applications of Khovanov homology in the context of tangles.