Parameterized complexity of quantum knot invariants

Clément Maria (1, 2)

1 DATASHAPE - Understanding the Shape of Data
2 Inria - Institut National de Recherche en Informatique et en Automatique

We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by 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)) 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 and the N th-coloured HOMFLYPT polynomial.

Mots clés: computational knot theory; parameterized complexity; quantum invariants; Theory of computation → Fixed parameter tractability; Theory of computation → Computational geometry; Mathematics of computing → Graphs and surfaces

Domaines: Géométrie algorithmique [cs.CG]; Topologie géométrique [math.GT]; Algèbres quantiques [math.QA]

Clément Maria. Parameterized complexity of quantum knot invariants. Symposium on Computational Geometry (SoCG), Jun 2021, Buffalo (NY), United States. ⟨10.4230/LIPIcs.SoCG.2021.53⟩. ⟨hal-02429767⟩