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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 As Get 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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