[Paper Review] Recurrence relation for HOMFLY polynomial and rational specializations

Rehana Ashraf, Barbu Berceanu|arXiv (Cornell University)|Mar 4, 2010

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper establishes a Fibonacci-type recurrence relation for the HOMFLY polynomial of closed braids, revealing that only three rational specializations exist: Alexander-Conway, Jones, and a newly identified polynomial. By leveraging this recurrence, the authors derive general and relative expansion formulas and rational generating functions, reducing computation to closures of simple braids, and prove algebraic independence of the three polynomials.

ABSTRACT

Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conway polynomial and the new polynomial, which reduce the computations to closure of simple braids, a subset of square free braids; HOMFLY polynomials of these simple braids are also computed. Algebraic independence of these three polynomials is proved.

Motivation & Objective

• To identify all rational specializations of the HOMFLY polynomial by transforming its skein relation into a Fibonacci-type recurrence.
• To develop general and relative expansion formulas for the Alexander-Conway polynomial and a new polynomial using the recurrence relation.
• To reduce the computation of HOMFLY polynomials to closures of simple braids—specifically, square-free braids—by exploiting the recurrence.
• To prove algebraic independence of the three polynomials: Alexander-Conway, Jones, and a newly discovered rational specialization.
• To provide rational generating functions and closed-form expressions for the Alexander-Conway and new polynomial via Laurent polynomial basis.

Proposed method

• Transform the HOMFLY skein relation into a multiple Fibonacci recurrence with parameters $(-ml, -l^2)$ for closed $n$-braids.
• Define a Laurent polynomial $C^a(s) = (-1)^a s^{a-1} + s^{1-a}$ as a basis for expansion formulas.
• Derive the general expansion formula: $\nabla_n(x_{i_1}^{a_1}\ldots x_{i_k}^{a_k}) = \left(\frac{s}{s^2+1}\right)^k \sum_{j_i \in \{0,1\}} \prod_{i=1}^k C^{a_i + j_i}(s) \cdot \nabla_n(x_{i_1}^{j_1}\ldots x_{i_k}^{j_k})$.
• Establish a relative expansion formula for links with $p$ subdiagrams: $\nabla_{\mathcal{L}}(k_1,\ldots,k_p) = \left(\frac{s}{s^2+1}\right)^p \sum_{j_i \in \{0,1\}} \prod_{i=1}^p C^{k_i + j_i}(s) \cdot \nabla_{\mathcal{L}}(j_1,\ldots,j_p)$.
• Use induction and asymptotic degree analysis to prove that certain Laurent polynomials in the recurrence must vanish, leading to algebraic independence.
• Apply the recurrence to construct a link $L_{p,n} = \widehat{x_1^{2p+1} \cdots x_n^{2p+1}}$ with distinct leading terms in $\nabla$, $V$, and $D$ polynomials to prove algebraic independence via contradiction on polynomial coefficients.

Experimental results

Research questions

• RQ1 What rational specializations of the HOMFLY polynomial satisfy a Fibonacci-type recurrence relation?
• RQ2 Can the Alexander-Conway and Jones polynomials be expressed via a general expansion formula using a canonical Laurent polynomial basis?
• RQ3 Is there a previously unknown rational specialization of the HOMFLY polynomial that arises from the recurrence structure?
• RQ4 Are the Alexander-Conway, Jones, and the new polynomial algebraically independent over $\mathbb{C}[s,s^{-1}]$?
• RQ5 Can the computation of HOMFLY polynomials be reduced to closures of simple braids using the recurrence?

Key findings

• There are exactly three rational specializations of the HOMFLY polynomial: Alexander-Conway, Jones, and a new one, derived from the recurrence's characteristic roots.
• The Alexander-Conway polynomial satisfies the multiple Fibonacci recurrence: $\nabla_n(a_1,\ldots,a_j+2,\ldots,a_k) = (s^{-1} - s)\nabla_n(a_1,\ldots,a_j+1,\ldots,a_k) + \nabla_n(a_1,\ldots,a_j,\ldots,a_k)$.
• The new polynomial arises when the characteristic equation has a double root $r''_1 = r''_2 = s$, distinct from the Jones and Alexander-Conway cases.
• The general expansion formula expresses the Alexander-Conway polynomial of any closed braid as a weighted sum over $2^k$ simple braid closures, with weights given by $C^{a_i + j_i}(s)$.
• The relative expansion formula allows computation of the Alexander-Conway polynomial for links with $p$ subdiagrams as a sum over $2^p$ configurations of the subdiagrams.
• The three polynomials—Alexander-Conway, Jones, and the new $D$-polynomial—are algebraically independent over $\mathbb{C}[s,s^{-1}]$, proven via asymptotic degree analysis of leading terms in a constructed family of links $L_{p,n}$.