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.
@InProceedings{maria:LIPIcs.SoCG.2021.53, author = {Maria, Cl\'{e}ment}, title = {{Parameterized Complexity of Quantum Knot Invariants}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {53:1--53:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.53}, URN = {urn:nbn:de:0030-drops-138527}, doi = {10.4230/LIPIcs.SoCG.2021.53}, annote = {Keywords: computational knot theory, parameterized complexity, quantum invariants} }
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