[Paper Review] The degree of the colored HOMFLY polynomial

Roland van der Veen|arXiv (Cornell University)|Dec 31, 2014

Advanced Combinatorial Mathematics12 references3 citations

TL;DR

This paper establishes new bounds on the degree of the colored HOMFLY polynomial in variables $a$ and $q$, generalizing Morton's bounds for the uncolored case. Using a reformulation of the MOY state sum via $q$-analogues of Ehrhart polynomials, the authors prove these bounds detect topological features such as boundary slopes of essential surfaces in the knot complement, and confirm a generalized HOMFLY slope conjecture for positive knots.

ABSTRACT

The colored HOMLFY polynomial is an important knot invariant depending on two variables $a$ and $q$. We give bounds on the degree in both $a$ and $q$ generalizing Morton's bounds \cite{Mo86} for the ordinary HOMFLY polynomial. Our bounds suggest that the degree detects certain incompressible surfaces in the knot complement and perhaps more generally features of the $SL(N)$ character varieties of the knot group. We formulate a precise conjecture along these lines generalizing the slope conjecture of Garoufalidis \cite{Ga11}. We prove our conjecture for all positive knots. Our technique is a reformulation of the MOY state sum \cite{MOY98} using $q$-analogues of Ehrhart polynomials. As a direct application we explicitly compute the $r$ coefficients of $r$-colored HOMFLY polynomial of any positive braid.

Motivation & Objective

• To establish explicit upper and lower bounds on the $a$- and $q$-degrees of the $r$-colored HOMFLY polynomial, generalizing Morton's bounds for $r=1$.
• To investigate the topological significance of the degree, particularly its relation to boundary slopes of essential surfaces in the knot complement.
• To formulate and prove a generalized HOMFLY slope conjecture, extending Garoufalidis's slope conjecture for the colored Jones polynomial.
• To develop a new algebraic framework using $q$-Ehrhart polynomials to re-express the MOY state sum for the HOMFLY polynomial.
• To compute the $r$-coefficients of the $r$-colored HOMFLY polynomial for positive braids via the new state sum formalism.

Proposed method

• Reformulate the MOY state sum using $q$-analogues of Ehrhart polynomials of order polytopes associated with elementary flow sequences.
• Define a new anti-symmetric evaluation of MOY graphs involving variables $a$, $q$, and $b$, where $b$ is shown to cancel out in the final evaluation.
• Apply $q$-Ehrhart reciprocity to relate the state sum to evaluations at $a = q^N$, $b = N$, recovering the $SL(N)$-invariant HOMFLY polynomial.
• Use the symmetry relation $P_r(L;q,a) = (-1)^r P_{r^t}(L;q^{-1},a)$ to derive a symmetric state sum for the colored HOMFLY polynomial.
• Leverage the fact that positive MOY diagrams admit only positive cycles, ensuring independence of the auxiliary parameter $b$.
• Prove that for positive knots, the degree bounds imply equality in the $q$-degree lower bound, leading to a proof of the generalized HOMFLY slope conjecture.

Experimental results

Research questions

• RQ1 Can the degree of the $r$-colored HOMFLY polynomial be bounded in terms of diagrammatic invariants like crossing signs and Seifert circle counts?
• RQ2 Does the growth rate of the $q$-degree, normalized by $r^{-2}$, detect boundary slopes of essential surfaces in the knot complement?
• RQ3 Is there a precise topological interpretation of the $q$-degree that generalizes the slope conjecture for the colored Jones polynomial?
• RQ4 Can the MOY state sum be re-expressed using $q$-Ehrhart polynomials of order polytopes to yield new structural insights?
• RQ5 What is the behavior of the $r$-coefficients of the $r$-colored HOMFLY polynomial for positive braids, and can they be explicitly computed?

Key findings

• The paper establishes sharp upper bounds on the $a$-degree: $\mathrm{maxdeg}_a P_r(D;a,q) \leq \frac{r}{2}(-c_+ + c_- + s_+ + s_-)$, matching Morton’s bound for $r=1$.
• For the $q$-degree, the upper bound is $\mathrm{maxdeg}_q P_r(D;a,q) \leq \frac{r}{2}(s_+ - s_- + c_+ + c_-(2r-1))$, with a matching lower bound for positive diagrams.
• For positive knots, the $q$-degree satisfies $\mathrm{maxdeg}_q P_r(K;a,q) = \frac{r}{2}(c_+ - s)$, implying $\mathcal{S}(K) = \{0\}$, thus confirming the HOMFLY slope conjecture.
• The $r$-coefficients of the $r$-colored HOMFLY polynomial for any positive braid are explicitly computable via the new state sum formalism.
• The reformulation using $q$-Ehrhart polynomials shows that the auxiliary parameter $b$ cancels in the final evaluation, validating the independence of the state sum from $b$.
• The HOMFLY slope conjecture $4\mathcal{S}(K) \subset \mathcal{B}(K)$ holds for all positive knots and is supported by examples of negative 2-braids with $\mathcal{S} = \{c_-/2\}$.