[Paper Review] Factorization of colored knot polynomials at roots of unity

А. Морозов|arXiv (Cornell University)|May 22, 2015

Geometric and Algebraic Topology53 references4 citations

TL;DR

The paper proposes a recursion relation for colored HOMFLY polynomials at roots of unity, showing that for symmetric representations [r] at q^{2m}=1, H_{r+m} = H_r H_m holds universally in A. This generalizes the special polynomial property H_r = (H_1)^r at q=1. The result implies non-trivial constraints on differential expansion coefficients, suggesting they are not free variables in knot space.

ABSTRACT

From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a generalization of the property H_r = (H_1)^r for special polynomials at q=1. We conjecture a natural generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from H_R at q=exp(i\pi/|R|), turns equal to the special polynomial with A substituted by A^|R|, provided R is a single-hook representations (e.g. arbitrary symmetric) -- what provides a q-A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots -- existence of such universal relations means that these variables are still not unconstrained.

Motivation & Objective

• To identify universal recursion structures in colored HOMFLY polynomials when q is a root of unity.
• To generalize the known property H_r = (H_1)^r (valid at q=1) to higher roots of unity for symmetric representations.
• To explore the behavior of Kashaev polynomials in relation to special polynomials and their A-duality for single-hook representations.
• To investigate constraints on differential expansion coefficients in the space of knots using these recursion identities.

Proposed method

• Analysis of a large class of knots to empirically identify a recursion identity: H_{r+m} = H_r H_m at q^{2m}=1 for symmetric representations [r].
• Conjecturing a generalization of the recursion to arbitrary representations R, with verification limited to torus knots.
• Studying the Kashaev polynomial as the HOMFLY polynomial evaluated at q = exp(iπ/|R|), where |R| denotes the size of the representation.
• Establishing a duality between the Kashaev polynomial and the special polynomial under the substitution A → A^|R| for single-hook representations.
• Deriving implications for differential expansions by analyzing the universal relations implied by the recursion and duality.

Experimental results

Research questions

• RQ1 Does the recursion H_{r+m} = H_r H_m hold for HOMFLY polynomials in symmetric representations at roots of unity q^{2m}=1?
• RQ2 Can this recursion be generalized to arbitrary representations R, particularly for torus knots?
• RQ3 How does the Kashaev polynomial relate to the special polynomial when R is a single-hook representation?
• RQ4 What constraints do the observed recursion and duality impose on the coefficients of differential expansions in knot invariants?
• RQ5 Are the differential expansion coefficients unconstrained, or do universal relations exist due to the algebraic structure at roots of unity?

Key findings

• At q^{2m}=1, HOMFLY polynomials in symmetric representations satisfy the universal recursion H_{r+m} = H_r H_m for any A.
• The recursion generalizes the known property H_r = (H_1)^r at q=1 to higher roots of unity, extending its validity beyond the special case.
• For single-hook representations (e.g., symmetric), the Kashaev polynomial at q = exp(iπ/|R|) equals the special polynomial with A replaced by A^{|R|}, establishing a q-A duality.
• The existence of such universal recursion and duality relations implies non-trivial constraints on the coefficients of differential expansions.
• These constraints suggest that the variables in the differential expansion are not independent, indicating a hidden algebraic structure in the space of knots.