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A parallel algorithm for the computation of the Jones polynomial.

Kasturi Barkataki1, Eleni Panagiotou1

  • 1School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85281.

Proceedings of the National Academy of Sciences of the United States of America
|April 21, 2026
PubMed
Summary
This summary is machine-generated.

We developed a parallel algorithm to quickly calculate the Jones polynomial, a key measure of knot complexity. This computational advance speeds up the analysis of entangled structures in science and engineering.

Keywords:
Jones polynomialentanglementknotsparallel programmingtopology

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Area of Science:

  • Computational topology
  • Applied mathematics
  • Complex systems analysis

Background:

  • Knot and link structures are fundamental in diverse scientific fields, with complexity correlating to function.
  • Efficient computation of topological invariants like the Jones polynomial is crucial for AI-driven discovery and mathematical proofs.
  • Current methods for computing the Jones polynomial are computationally intensive (#P-hard).

Purpose of the Study:

  • Introduce a novel parallel algorithm for the exact computation of the Jones polynomial.
  • Address the computational bottleneck in analyzing topological complexity.
  • Demonstrate the algorithm's efficiency and applicability.

Main Methods:

  • Development of a parallel algorithm for computing the Jones polynomial.
  • Application to open and closed simple curves in 3-space.
  • Validation using knot theory examples and molecular dynamics simulations of polymer melts.

Main Results:

  • The parallel algorithm achieves an exponential reduction in computational time relative to the number of processors.
  • Successfully computed the Jones polynomial for complex systems, including polymer melts.
  • Demonstrated significant performance advantages over existing methods.

Conclusions:

  • The new parallel algorithm offers a significant speedup for computing the Jones polynomial.
  • This method facilitates the analysis of complex topological structures in various scientific domains.
  • The approach is general and potentially applicable to other topological invariants.