Efficient Computation of the Kauffman Bracket

TL;DR

This paper presents a new method to compute the Kauffman bracket more efficiently by bounding the computational complexity in terms of the link's crossing number, reducing the problem to manageable linear combinations and boundary points.

Contribution

It introduces bounds on the Kauffman bracket computation, showing it can be done in polynomial time times an exponential of the square root of the crossing number.

Findings

The image of a tangle in the skein module is a linear combination of O(2^g) basis elements.

Each coefficient in the combination has at most n nonzero terms with integer coefficients.

The computation time is polynomial in n times 2^{C√n}.

Abstract

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket skein module is a linear combination of $O(2^g)$ basis elements, with each coefficient a polynomial with at most $n$ nonzero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by $C\sqrt{n}$ for some $C.$ From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in $n$ times $2^{C\sqrt{n}}.$