A fast algorithm for the Hecke representation of the braid group, and applications to the computation of the HOMFLY-PT polynomial and the search for interesting braids

TL;DR

This paper introduces a fast algorithm for the Hecke representation of the braid group, enabling efficient computation of the HOMFLY-PT polynomial and discovery of non-trivial braids with trivial Hecke representations, advancing computational knot theory.

Contribution

The paper presents a novel, efficient algorithm for the Hecke representation of braid groups and applies it to compute knot invariants and identify interesting braids, including non-faithful cases.

Findings

Developed a fast Hecke representation algorithm for braid groups.

Successfully computed the HOMFLY-PT polynomial using the new algorithm.

Discovered non-trivial braids with trivial Hecke representations, including non-faithfulness of B5 with Z/2Z coefficients.

Abstract

Knot theory is an active field of mathematics, in which combinatorial and computational methods play an important role. One side of computational knot theory, that has gained interest in recent years, both for complexity analysis and practical algorithms, is quantum topology and the computation of topological invariants issued from the theory. In this article, we leverage the rigidity brought by the representation-theoretic origins of the quantum invariants for algorithmic purposes. We do so by exploiting braids and the algebraic properties of the braid group to describe, analyze, and implement a fast algorithm to compute the Hecke representation of the braid group. We apply this construction to design a parameterized algorithm to compute the HOMFLY-PT polynomial of knots, and demonstrate its interest experimentally. Finally, we combine our fast Hecke representation algorithm with Garside theory, to implement a reservoir sampling search and find non-trivial braids with trivial Hecke representations with coefficients in $\mathbb{Z}/p\mathbb{Z}$. We find several such braids, in particular proving that the Hecke representation of $B_5$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients is non-faithful, a previously unknown fact.