Parameterized Complexity of Quantum Knot Invariants

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 2 cwpoly(n)) ∈ NO( √ n) time algorithm to compute any ReshetikhinTuraev 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. 2012 ACM Subject Classification Theory of computation → Fixed parameter tractability; Theory of computation → Computational geometry; Mathematics of computing → Graphs and surfaces