On computational complexity of Khovanov homology

TL;DR

This paper explores the computational complexity of Khovanov homology, demonstrating polynomial time algorithms for certain braid classes and establishing bounds on homology ranks, thus advancing understanding of its algorithmic aspects.

Contribution

It introduces polynomial time algorithms for Khovanov homology of 3-braids and provides bounds on homology ranks, highlighting complexity distinctions.

Findings

Polynomial time algorithm for 3-braids Khovanov homology.

Exponential time of Bar-Natan's scanning algorithm on simple 3-braids.

Bounds on ranks of Khovanov homology groups.

Abstract

Computing the Jones polynomial of general link diagrams is known to be $\#$P-hard, while restricting the computation to braid closures on fixed number of strands allows for a polynomial time algorithm. We investigate polynomial time algorithms for Khovanov homology of braids and show that for $3$-braids there is one. In contrast, we show that Bar-Natan's scanning algorithm runs in exponential time when restricted to simple classes of $3$-braids. For more general braids, we obtain that a variation of the scanning algorithm computes the Khovanov homology for a bounded set of homological degrees in polynomial time. We also prove upper and lower bounds on the ranks of Khovanov homology groups.