A parallel algorithm for the computation of the Jones polynomial

TL;DR

This paper introduces a parallel algorithm for the exact computation of the Jones polynomial, significantly reducing computation time for knots and entangled systems, with applications in biology, engineering, and data analysis.

Contribution

It presents the first parallel algorithm for computing the Jones polynomial, enabling exponential speedup depending on the number of processors.

Findings

Reduces computational time exponentially with more processors

Successfully applied to knots and polymer systems from simulations

General method applicable to other invariants and complexity measures

Abstract

Knots, links and entangled filaments appear in many physical systems of interest in biology and engineering. Classifying knots and measuring entanglement is of interest both for advancing knot theory, as well as for analyzing large data that become available through experiments or Artificial Intelligence. In this context, the efficient computation of topological invariants and other metrics of entanglement becomes an urgent issue. The computation of common measures of topological complexity, such as the Jones polynomial, is #P-hard and of exponential time on the number of crossings in a knot(oid) (link(oid)) diagram. In this paper, we introduce the first parallel algorithm for the exact computation of the Jones polynomial for (collections of) both open and closed simple curves in 3-space. This algorithm enables the reduction of the computational time by an exponential factor depending on the number of processors. We demonstrate the advantage of this algorithm by applying it to knots, as well as to systems of linear polymers in a melt obtained from molecular dynamics simulations. The method is general and could be applied to other invariants and measures of complexity.