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Determinantal representation of colored HOMFLY for double braids
Abstract
Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations [rs] as determinants of r× r and s× 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.
Related
- Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials
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BibTeX
@misc{kononov2026determinantal,
title = {{Determinantal representation of colored HOMFLY for double braids}},
author = {Ya. Kononov and A. Morozov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09437},
eprint = {2609.09437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09437},
}
RIS
TY - UNPB
TI - Determinantal representation of colored HOMFLY for double braids
AU - Ya. Kononov
AU - A. Morozov
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09437
UR - https://arxiv.org/abs/2609.09437
AB - 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× r and s× 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.
ER -