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

2025-12-05

AI Summary

This paper addresses the efficient computation of Hecke algebra representations of braid groups and their application to knot invariants. Methodologically, it introduces a novel algorithm integrating the Garside structure of braid groups with Hecke algebra representations, leveraging algebraic properties of braids to construct a parametrized computational framework and employing reservoir sampling to enable large-scale combinatorial search. The main contributions are threefold: (1) the first proof that the Hecke representation of the braid group (B_5) is unfaithful over (mathbb{Z}/2mathbb{Z}); (2) the systematic discovery of multiple nontrivial braids whose Hecke representations are trivial; and (3) a substantial improvement in the computational efficiency of HOMFLY-PT polynomial evaluation, thereby demonstrating the algorithm’s dual efficacy in both invariant computation and structural exploration of Hecke representations.

Technology Category

• Search and Optimization: Heuristic Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $mathbb{Z}/pmathbb{Z}$. We find several such braids, in particular proving that the Hecke representation of $B_5$ with $mathbb{Z}/2mathbb{Z}$ coefficients is non-faithful, a previously unknown fact.

Problem

• Develop a fast algorithm for the Hecke representation of the braid group
• Apply the algorithm to compute the HOMFLY-PT polynomial of knots efficiently
• Search for non-trivial braids with trivial Hecke representations over finite fields

Innovation

• Fast algorithm for Hecke representation of braid group
• Parameterized algorithm for HOMFLY-PT polynomial computation
• Combining Hecke algorithm with Garside theory for search