The HOMFLY-PT polynomial is fixed-parameter tractable

TL;DR

This paper proves that computing the HOMFLY-PT polynomial, a key knot invariant, is fixed-parameter tractable with respect to treewidth, and provides the first sub-exponential algorithm for arbitrary links.

Contribution

It establishes fixed-parameter tractability of HOMFLY-PT polynomial computation in terms of treewidth and introduces the first sub-exponential algorithm for general links.

Findings

HOMFLY-PT polynomial computation is fixed-parameter tractable.

First sub-exponential algorithm for computing HOMFLY-PT for arbitrary links.

Extends fixed-parameter tractability results from Jones to HOMFLY-PT polynomial.

Abstract

Many polynomial invariants of knots and links, including the Jones and HOMFLY-PT polynomials, are widely used in practice but #P-hard to compute. It was shown by Makowsky in 2001 that computing the Jones polynomial is fixed-parameter tractable in the treewidth of the link diagram, but the parameterised complexity of the more powerful HOMFLY-PT polynomial remained an open problem. Here we show that computing HOMFLY-PT is fixed-parameter tractable in the treewidth, and we give the first sub-exponential time algorithm to compute it for arbitrary links.