Computing Finite Type Invariants Efficiently

TL;DR

This paper introduces an efficient algorithm for computing finite type invariants of knots, significantly reducing the computational complexity from previous methods by utilizing a look-up table for subdiagrams.

Contribution

The paper presents a novel algorithm that computes finite type invariants more efficiently, reducing the time complexity from ilde{O}(n^k) to ilde{O}(n^{ ext{ceil}(k/2)}).

Findings

Algorithm computes invariants faster than previous methods.

Time complexity improved from ilde{O}(n^k) to ilde{O}(n^{ ext{ceil}(k/2)}).

Applicable to knots with n crossings for finite type invariants of type k.

Abstract

We describe an efficient algorithm to compute finite type invariants of type $k$ by first creating, for a given knot $K$ with $n$ crossings, a look-up table for all subdiagrams of $K$ of size $\lceil \frac{k}{2}\rceil$ indexed by dyadic intervals in $[0,2n-1]$. Using this algorithm, any such finite type invariant can be computed on an $n$-crossing knot in time $\tilde{O}( n^{\lceil \frac{k}{2}\rceil})$, a lot faster than the previously best published bound of $\tilde{O} (n^k)$.