Towards effective topological field theory for knots

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 [25] corresponds to the case m=2, and our generalization sheds additional light on the structure of those mysterious formulas. Explicit expressions are now combined from Racah matrices of the type R⊗R⊗R¯⟶R¯ and mixing matrices in the sectors R⊗3⟶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).