Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials
Abstract
We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in ℤ[A±1,q±1]. These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.
Related
- Cyclotomic expansions of colored SU(n) invariants of two-strand torus knots and Bailey transforms
- Determinantal representation of colored HOMFLY for double braids
- Cyclotomic factorization and fixed-point insertion in colored permutation groups
- Post-critically finite polynomials over cyclotomic closures of number fields
- Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion
BibTeX
@misc{fang2026rectangular,
title = {{Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials}},
author = {Honghuai Fang and Tian Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00303},
eprint = {2610.00303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00303},
}
RIS
TY - UNPB
TI - Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials
AU - Honghuai Fang
AU - Tian Zhou
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00303
UR - https://arxiv.org/abs/2610.00303
AB - We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in ℤ[A^±1,q^±1]. These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.
ER -