Towards effective topological field theory for knots

TL;DR

This paper generalizes the construction of colored knot polynomials for complex braid structures, providing explicit formulas and composition rules that enhance the understanding of topological invariants in knot theory.

Contribution

It introduces a generalized framework for constructing knot polynomials using Racah and mixing matrices for arbitrary m-strand braids, extending previous models.

Findings

Explicit formulas for knot polynomials involving Racah matrices.

New composition rules for gluing braid blocks.

Enhanced understanding of the structure of knot invariants.

Abstract

Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371 corresponds to the case m=2, and our generalizations sheds additional light on the structure of those mysterious formulas. Explicit expressions are now combined from Racah matrices of the type $R\otimes R\otimes\bar R\longrightarrow \bar R$ and mixing matrices in the sectors $R^{\otimes 3}\longrightarrow Q$. Further extension is provided by composition rules, allowing to glue two blocks, connected by an m-strand braid (they generalize the product formula for ordinary composite knots with m=1).