Towards differential expansion for generic representations
Abstract
We report the first results of phenomenological analysis, based on extended set of explicitly calculated colored HOMFLY-PT polynomials in \cite{knotebook}. They concern differential expansion (DE), which separates the dependencies on the knot cal K and the representation R. %of colored HOMFLY-PT polynomials. DE is well understood for symmetric and rectangular representations, while for non-rectangular R even the knot-independent basis ZRX of DE is not known. We concentrate on two issues: the possible derivation of DE and on improved characteristic of its defect. We discuss how evolution families -- twist knots, double braids and 3-pretzels -- can be used to find DE by a peculiar triangular transformation of evolution coefficients. Once ZRX are found for particular family, they are expected to stay basically the same for arbitrary knots -- modulo controllable reductions, characterized by the defect of DE, which puts knots into different universality classes. Contrary to original expectation, these classes appear labeled not by a single number δcal K, but by a whole non-decreasing sequence {kj}, which is squeezed between two staircases, entier ((j−1)/(δ+1))≤ kj≤ entier ((j−1)/(deg Δ)), where the upper bound involves Alexander polynomial Δ(q2) and follows from the continuation of the Melvin--Morton--Rozansky-Kashaev limit to A=q−m with m>0, when the limit gives a {\it positive} power of Alexander. The old assumption δ+1=deg Δ is oversimplified and fails for 19 non-alternating knots with 11 and 12 crossings. Last, we list the differences of non-rectangularly colored HOMFLY for various mutant pairs to demonstrate their universality (knot-independence) and nice consistency with the spirit of DE.
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BibTeX
@misc{kononov2026towards,
title = {{Towards differential expansion for generic representations}},
author = {Ya. Kononov and A. Morozov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28531},
eprint = {2610.28531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28531},
}
RIS
TY - UNPB
TI - Towards differential expansion for generic representations
AU - Ya. Kononov
AU - A. Morozov
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.28531
UR - https://arxiv.org/abs/2610.28531
AB - We report the first results of phenomenological analysis, based on extended set of explicitly calculated colored HOMFLY-PT polynomials in \cite{knotebook}. They concern differential expansion (DE), which separates the dependencies on the knot cal K and the representation R. %of colored HOMFLY-PT polynomials. DE is well understood for symmetric and rectangular representations, while for non-rectangular R even the knot-independent basis Z_R^X of DE is not known. We concentrate on two issues: the possible derivation of DE and on improved characteristic of its defect. We discuss how evolution families -- twist knots, double braids and 3-pretzels -- can be used to find DE by a peculiar triangular transformation of evolution coefficients. Once Z^R_X are found for particular family, they are expected to stay basically the same for arbitrary knots -- modulo controllable reductions, characterized by the defect of DE, which puts knots into different universality classes. Contrary to original expectation, these classes appear labeled not by a single number δ^cal K, but by a whole non-decreasing sequence {k_j}, which is squeezed between two staircases, entier ((j−1)/(δ+1))≤ k_j≤ entier ((j−1)/(deg Δ)), where the upper bound involves Alexander polynomial Δ(q^2) and follows from the continuation of the Melvin--Morton--Rozansky-Kashaev limit to A=q^−m with m>0, when the limit gives a {\it positive} power of Alexander. The old assumption δ+1=deg Δ is oversimplified and fails for 19 non-alternating knots with 11 and 12 crossings. Last, we list the differences of non-rectangularly colored HOMFLY for various mutant pairs to demonstrate their universality (knot-independence) and nice consistency with the spirit of DE.
ER -