Fixed-parameter tractable computation of Reshetikhin--Turaev knot polynomials via tensor networks

Abstract

We give a thorough analysis of the time complexity of computing Reshetikhin--Turaev knot polynomials via tensor contractions on the associated tensor networks, showing that the computation is fixed-parameter tractable with respect to a parameter at most linear in the tree-width of the input knot diagram. When combined with existing approximation algorithms for tree decomposition, this recovers the sub-exponential bound $e^{O(\sqrt{n})}$ for the time complexity of computing any Reshetikhin--Turaev knot polynomial. We accompany this paper with an implementation of such an algorithm in SnapPy, which computes any Reshetikhin--Turaev knot polynomial given its $R$-matrix and ribbon element.