Towards effective topological field theory for knots

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 )