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Parameterized Complexity of Quantum Knot Invariants

Author Clément Maria

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Author Details

Clément Maria
  • INRIA Sophia Antipolis-Méditerranée, France

Funding

  • Maria, Clément: This work has been partially supported by the ANR project ANR-20-CE48-0007-01 (AlgoKnot).

Cite AsGet BibTex

Clément Maria. Parameterized Complexity of Quantum Knot Invariants. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 53:1-53:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.53

Abstract

We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by planar diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a O(N^{3/2 cw} poly(n)) ∈ N^O(√n) time algorithm to compute any Reshetikhin-Turaev invariant - derived from a simple Lie algebra 𝔤 - of a link presented by a planar diagram with n crossings and carving-width cw, and whose components are coloured with 𝔤-modules of dimension at most N. For example, this includes the N^{th}-coloured Jones polynomial.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graphs and surfaces
Keywords
  • computational knot theory
  • parameterized complexity
  • quantum invariants

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