Braids and combinatorial knot Floer homology

TL;DR

This paper introduces a braid-based combinatorial method for computing knot Floer homology, providing an explicit algorithm that is often faster than previous grid diagram approaches.

Contribution

It develops a braid-theoretic framework and an explicit algorithm for knot Floer homology computation, improving efficiency over prior methods.

Findings

Algorithm significantly faster than grid diagram methods in many cases

Provides explicit bounds on computational complexity

Establishes a braid-based approach for combinatorial knot Floer homology

Abstract

We present a braid-theoretic approach to combinatorially computing knot Floer homology. To a knot or link K, which is braided about the standard disk open book decomposition for (S^3,\xi_std), we associate a corresponding multi-pointed nice Heegaard diagram. We then describe an explicit algorithm for computing the associated knot Floer homology groups. We compute explicit bounds for the computational complexity of our algorithm and demonstrate that, in many cases, it is significantly faster than the previous approach using grid diagrams.