The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming

TL;DR

This paper presents a polynomial time algorithm for computing the first coefficients of the Homflypt and Kauffman polynomials, resolving a long-standing computational complexity question in knot theory.

Contribution

It provides the first published polynomial time algorithm for these coefficients, based on Vertigan's unpublished discovery from 1992.

Findings

Algorithm computes first coefficients efficiently in polynomial time.

Establishes polynomial complexity of these polynomial invariants.

Resolves a previously open problem in computational knot theory.

Abstract

We describe the polynomial time complexity algorithm for computing first coefficients of the skein (Homflypt) and Kauffman polynomial invariants of links, discovered by D.Vertigan in 1992 but never published.