[Paper Review] New formulas for the Jones polynomial of a rational link

Yuanan Diao, Gábor Hetyei|arXiv (Cornell University)|Mar 22, 2026

Advanced Combinatorial Mathematics0 citations

TL;DR

The paper derives new formulas for the Jones and Kauffman bracket polynomials of rational links in standard (not necessarily alternating) diagrams, using colored Tutte polynomials and colored tensor products, and provides a finite automaton for crossing signs to compute writhe.

ABSTRACT

We derive new formulas for the Jones polynomial and the Kauffman bracket polynomial of a rational link represented by a standard diagram that is not necessarily alternating. These formulas generalize the results of Qazaqzeh, Yasein, and Abu-Qamar for the Tutte polynomial of the Tait graph of an alternating diagram of a rational link, as well as the matrix formulas of Lawrence and Rosenstein for the Jones polynomial of a rational link. Our approach uses the colored version of Brylawski's tensor product formula for Tutte polynomials of colored graphs, due to Diao, Hetyei, and Hinson. Furthermore, generalizing the formulas of Qazaqzeh, Yasein, and Abu-Qamar, we present a finite automaton that computes the crossing signs, thereby enabling the calculation of the writhe of a standard diagram of a rational link.

Motivation & Objective

• Generalize Jones polynomial computation to rational links represented by standard (non-alternating) diagrams.
• Extend Brylawski’s tensor product framework to colored graphs to handle augmented path graphs and their duals.
• Relate the Tutte invariants of the Tait graph to the Kauffman bracket and Jones polynomial via colored substitutions.
• Provide efficient matrix and continued fraction formulations for computing the Jones polynomial of rational links.

Proposed method

• Represent the Tait graph of a rational link as a colored tensor product of a core graph with augmented path graphs and their duals.
• Compute colored Tutte polynomials for augmented path graphs and their duals (and their pointed versions).
• Compute colored Tutte polynomials for core graphs and assemble the full Jones-related polynomials via tensor-product substitutions (Theorem 1.6).
• Derive variants and special cases including matrix-factor formulations (Lawrence–Rosenstein type) and continued fraction connections.
• Introduce a finite automaton approach to determine crossing signs and writhe for standard diagrams (Section 10).
• Relate the Kauffman bracket to the Jones polynomial through substitutions in the Tutte framework (Theorem 1.7 and Corollary 1.8).

Experimental results

Research questions

• RQ1 How can the Jones polynomial of a rational link be computed from a non-alternating standard diagram?
• RQ2 Can colored tensor-product generalizations of Brylawski’s tensor product formula efficiently yield Tutte invariants for augmented path graphs and their duals?
• RQ3 What is a closed-form or matrix-product expression for the Kauffman bracket and Jones polynomial of rational links via core graphs and augmented paths?
• RQ4 How can one compute writhe in this framework, and can automata provide a general method for crossing-sign computation?
• RQ5 Do these methods recover known results for alternating rational links and relate to continued fractions and Lickorish–Millett type formulations?

Key findings

• The Tait graph of a rational link diagram in standard form decomposes as a colored tensor product of a core graph with augmented path graphs and their duals (Theorem 2.6).
• The colored Tutte invariants of augmented path graphs and their duals have explicit closed forms and can be computed via simple tiling interpretations (Theorem 4.1).
• The colored Tutte polynomials of the core graphs satisfy recurrence relations that lead to matrix product representations and generalized continued fractions (Section 4 and 5).
• Substitutions linking colored Tutte polynomials to the Kauffman bracket yield the Jones polynomial after adjusting for writhe (Theorem 1.7 and Corollary 1.8).
• A finite automaton approach is provided to compute crossing signs and writhe for standard rational link diagrams (Section 10).
• Special cases recover matrix-factor formulations akin to Lawrence and Rosenstein and extend previous alternating-link results to non-alternating rational links (Sections 5–9).