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A parallel algorithm for the computation of the Jones polynomial

Kasturi Barkataki, Eleni Panagiotou

TL;DR

This paper introduces the first parallel algorithm for the exact computation of the Jones polynomial for (collections of) both open and closed simple curves in 3-space and enables the reduction of the computational time by an exponential factor depending on the number of processors.

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.

A parallel algorithm for the computation of the Jones polynomial

TL;DR

This paper introduces the first parallel algorithm for the exact computation of the Jones polynomial for (collections of) both open and closed simple curves in 3-space and enables the reduction of the computational time by an exponential factor depending on the number of processors.

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.

Paper Structure

This paper contains 12 sections, 9 theorems, 24 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Jones polynomial
  3. A parallel algorithm for the Jones polynomial
  4. Jones polynomial of a knot in terms of the Jones polynomial of its constituent linkoids
  5. Conclusion
  6. Acknowledgement
  7. Declarations
  8. Appendix
  9. Distinct state permutations and segment cycles in linkoids
  10. Distinct state permutations of a linkoid diagram
  11. Number of segment cycles in a linkoid state
  12. The parallel algorithm and treewidth parameter of link(oid) diagrams

Key Result

Theorem 3.1

Let $L={\cup_{\sigma}}_{i=1}^{2^m} L_i$ denote a linkoid $L$ obtained by the gluing of $2^m$ disjoint linkoid diagrams, $L_i$, where $1\leq i \leq 2^m$ and $m\in \mathbb{N}$, with gluing permutation $\sigma$. The Jones polynomial of $L$ is $f_L=(-A^3)^{-Wr(L)}\langle L \rangle$, where the bracket po where ${\cal{S}}$ is a state of $L$ expressed as a gluing of states of the constituent linkoids. Mo

Figures (4)

  • Figure 1: Omega moves (Reidemeister moves $\Omega_1,\Omega_2,\Omega_3$) and forbidden moves ($\Phi_+, \Phi_-$) on linkoid diagrams.
  • Figure 2: A diagram of a link(oid) can be seen as the gluing of two linkoid diagrams. (Left) A diagram of a pure knotoid, $K$, and the axis along which it is to be split with the starting point of $L_1$ denoted $\star$. (Right) The two subdivided linkoid pieces, $L_1$ and $L_2$, with strand permutations $(1 \quad 2)(3 \quad 4)$ and $(5 \quad 6)(7 \quad 8)$, respectively, on the labelled endpoints obtained upon the splitting. The closure permutation $(4 \quad 5)(8 \quad 3)(6 \quad 1)$ on $L_1 \cup L_2$ retrieves $K$. The Jones polynomial computation for each of the two components is done in parallel and the results are combined, also in parallel, to give the Jones polynomial of the knot.
  • Figure 3: Computational time (in seconds) of the Jones polynomial as a function of the number of crossings in a diagram using the serial algorithm and the parallel algorithm, Algorithm \ref{['alg:parallel_jones']}, with 2 subdivisions and 2 processors.
  • Figure 4: Computational time (in hours) of the Jones polynomial of linear chains in a polymer melt as a function of molecular weight using the serial algorithm and the parallel algorithm, Algorithm \ref{['alg:parallel_jones']}, with 2 subdivisions and 2 processors.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.1
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • proof
  • ...and 25 more