Colored HOMFLY polynomials of knots presented as double fat diagrams

TL;DR

The paper develops a robust, diagrammatic approach (double fat diagrams) to compute $[21]$-colored HOMFLY polynomials by encoding knot structure in four-strand braids with Racah matrices $S,ar S$ and braiding matrices $T$. It extends the framework beyond rectangular representations to non-rectangular ones, enabling explicit calculation and mutation analysis for knots and pretzel families, and provides extensive verification against known results and evolution-based predictions. A key finding is that $[21]$-colored HOMFLY can distinguish certain mutants (e.g., Kinoshita–Terasaka vs Conway and many pretzel mutants) while remaining blind to others, with differences constrained by universal factors and differential expansions. The work delivers not only wide-ranging explicit polynomials (notably for $[21]$) but also a systematic sign prescription and a suite of checks (special/Alexander/Jones, differential expansion, quasiclassical invariants) that bolster the framework, suggesting a powerful avenue to explore knot invariants in arbitrary representations and to probe topological field theory interpretations of knot diagrams.

Abstract

Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.

Figures (5)

• Figure 1: Knot drawn as double fat diagram
• Figure 2: knot equivalence
• Figure 3: (a) Kinoshita-Terasaka (b) Conway
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