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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

Authors Clément Maria , Hoel Queffelec

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LIPIcs.SoCG.2026.76.pdf
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ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Fixed parameter tractability
Keywords
  • Hecke representation of the braid group
  • parameterized algorithm
  • HOMFLY-PT polynomial of knots
  • reservoir sampling
  • faithfulness of Hecke representation

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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 ℤ/pℤ. We find explicitly several such braids, for the 4-strand and 5-strand braid groups.

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Clément Maria and Hoel Queffelec. 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. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 76:1-76:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.76

Author Details

Clément Maria
  • Inria d'Université Côte d'Azur, Valbonne, France
Hoel Queffelec
  • France-Australia Mathematical Sciences and Interactions ANU - CNRS International Research Laboratory, Canberra, Australia

Funding

  • Maria, Clément: Partially supported by the ANR project ANR-20-CE48-0007 (AlgoKnot).

Acknowledgements

We would like to thank Asilata Bapat, Stepan Orevkov - who pointed us to Morton’s 1985 implementation [H. Morton, 1985] of the HOMFLY-PT algorithm - , Emmanuel Wagner and Oded Yacobi for helpful discussions. We would also like to thank the anonymous referees for the detailed comments and useful feedback. Some experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000.fr).

Supplementary Materials

References

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