[Paper Review] An Efficient Algorithm to Compute the Colored Jones Polynomial

Mustafa Hajij, Jesse Levitt|arXiv (Cornell University)|Apr 21, 2018

Geometric and Algebraic Topology26 references3 citations

TL;DR

This paper presents a highly optimized classical algorithm to compute the colored Jones polynomial (CJP) of any knot using walks along a braid model, leveraging a refined q-Weyl algebra framework and a novel Duplicate Reduction Algorithm (DRA). The method achieves up to an order of magnitude speedup over existing implementations, including the KnotTheory package, by minimizing redundant word evaluations and exploiting modern Mathematica features for efficient symbolic computation.

ABSTRACT

The colored Jones polynomial is a knot invariant that plays a central role in low dimensional topology. We give a simple and an efficient algorithm to compute the colored Jones polynomial of any knot. Our algorithm utilizes the walks along a braid model of the colored Jones polynomial that was refined by Armond from the work of Huynh and Lê. The walk model gives rise to ordered words in a $q$-Weyl algebra which we address and study from multiple perspectives. We provide a highly optimized Mathematica implementation that exploits the modern features of the software. We include a performance analysis for the running time of our algorithm. Our implementation of the algorithm shows that our method usually runs in faster time than the existing state-of the-art method by an order of magnitude.

Motivation & Objective

• To develop a fast, general-purpose classical algorithm for computing the colored Jones polynomial (CJP) of any knot.
• To overcome the limitations of prior methods—such as diagrammatic complexity and memory overflow—by using a walk-based model on braids.
• To optimize the computation of CJP by minimizing the number of required walks using a structural theorem and a new Duplicate Reduction Algorithm (DRA).
• To provide a highly efficient, fully functional Mathematica implementation that leverages modern software features for symbolic computation.
• To demonstrate significant performance gains over state-of-the-art implementations, particularly for higher color representations (N ≥ 9) and knots with more crossings.

Proposed method

• The algorithm models the CJP computation via walks on a braid representation of the knot, translating each walk into an ordered word in a q-Weyl algebra.
• It introduces a standard word form to uniquely represent and simplify algebraic expressions in the q-Weyl algebra, reducing redundancy.
• A key innovation is the Duplicate Reduction Algorithm (DRA), which minimizes the number of distinct walks needed to compute the CJP by eliminating equivalent or redundant paths.
• The method uses a structural theorem on braid walks to guide the DRA and reduce the total number of evaluations required.
• The algorithm is implemented in Mathematica, utilizing modern features such as pattern matching, memoization, and symbolic simplification for high performance.
• The CJP is computed by evaluating all standard words and summing their contributions as Laurent polynomials in q.

Experimental results

Research questions

• RQ1 Can a classical algorithm be designed to compute the colored Jones polynomial efficiently for all knots, regardless of crossing number or coloring?
• RQ2 How can the number of walks in the braid model be minimized without losing computational accuracy?
• RQ3 To what extent can modern symbolic computation tools like Mathematica be leveraged to optimize the evaluation of non-commutative algebraic expressions arising from the CJP?
• RQ4 What is the asymptotic growth of the number of required walks with respect to the number of crossings, and can it be bounded?
• RQ5 How does the performance of the proposed algorithm compare to existing implementations, especially for higher color representations and complex knots?

Key findings

• The proposed algorithm runs approximately one order of magnitude faster than the KnotTheory package’s REngine method across the first 28 knots in the knot table.
• The Duplicate Reduction Algorithm (DRA) reduces the number of required walks by eliminating redundant path evaluations, significantly lowering computational load.
• For knots with 9 crossings, the algorithm completes computations that previously caused memory overflow in the KnotTheory package, demonstrating improved scalability.
• The average number of simple walks grows as O(k²) with respect to the number of crossings k, suggesting a manageable growth rate for practical computation.
• The algorithm’s running time scales linearly with the number of walks, and performance analysis confirms consistent speedups across different colors (N = 2 to 7).
• The implementation remains stable and does not experience memory overflow during testing, unlike the KnotTheory package under similar workloads.