Colored HOMFLY polynomials of knots presented as double fat diagrams

Understand

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.