Year: 2026
Type: preprint
Abstract: Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations \([r^s]\) as determinants of \(r\times r\) and \(s\times s\) matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda \(τ\)-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.
Source: arXiv (Cornell University)
Authors: Kononov, Ya., Morozov, A.
Institution: Institute for Theoretical and Experimental Physics
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212133136