[Paper Review] The non-commutative $A$-polynomial of twist knots

Stavros Garoufalidis, Xinyu Sun|ArXiv.org|Feb 27, 2008

Geometric and Algebraic Topology22 references4 citations

TL;DR

This paper introduces a multivariable creative telescoping method to compute the non-commutative $A$-polynomial of twist knots, a quantum invariant encoding the $q$-difference equation satisfied by their colored Jones polynomials. The method efficiently derives minimal-order $q$-difference equations, enabling explicit computation of the non-commutative $A$-polynomial for twist knots with $-8$ to $11$ crossings, providing new verification of the AJ-Conjecture for these knots.

ABSTRACT

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative $A$-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form $J(n)=\sum_k c(n,k) \hatJ (k)$ given a recursion relation for $(\hatJ(n))$ a the hypergeometric kernel $c(n,k)$. As an application of our method, we explicitly compute the non-commutative $A$-polynomial for twist knots with -8 and 11 crossings. The non-commutative $A$-polynomial of a knot encodes the monic, linear, minimal order $q$-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to $q=1$ is conjectured to be the better-known $A$-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the $A$-polynomial is harder to compute and already unknown for some knots with 12 crossings.

Motivation & Objective

• To develop a multivariable creative telescoping method for computing linear recursions of sums involving hypergeometric kernels and recursively defined sequences.
• To apply this method to compute the non-commutative $A$-polynomial for twist knots with $-8$ to $11$ crossings, a significant class of hyperbolic knots.
• To provide explicit, minimal-order $q$-difference equations satisfied by the colored Jones polynomials of these knots, thereby offering new evidence for the AJ-Conjecture.
• To overcome the computational intractability of standard methods in deriving the non-commutative $A$-polynomial for knots with more than 12 crossings.

Proposed method

• The method uses a multivariable generalization of the WZ and creative telescoping algorithms to derive linear recursions for sums of the form $J(n) = \sum_k c(n,k) \hat{J}(k)$, given a recursion for $\hat{J}(n)$.
• It introduces the concept of a 'multi-certificate' to handle multiple summation indices and reduce the problem to solving a system of linear equations.
• The approach leverages the $q$-holonomicity of the colored Jones function and constructs a $q$-difference operator annihilating the sequence $J(n)$.
• The method is implemented algorithmically using symbolic computation, with the key step being the solution of a linear system derived from the multi-certificate condition.
• The non-commutative $A$-polynomial is extracted as the minimal-order monic $q$-difference operator annihilating the colored Jones sequence.
• The method is applied to twist knots by expressing their Jones polynomials as sums over hypergeometric terms and applying the recursive reduction.

Experimental results

Research questions

• RQ1 Can a multivariable creative telescoping method be constructed to compute $q$-difference equations for sequences defined as sums of hypergeometric terms?
• RQ2 What is the non-commutative $A$-polynomial of twist knots with $-8$ to $11$ crossings, and does it satisfy the minimality and monicity conditions required by the AJ-Conjecture?
• RQ3 Can this method produce the non-commutative $A$-polynomial efficiently and in minimal order, where standard elimination methods fail?
• RQ4 Is there a structural recursion in the number of twists for the non-commutative $A$-polynomial of twist knots?

Key findings

• The non-commutative $A$-polynomial of the twist knot with $p = -3$ is explicitly computed as a degree-6 polynomial in $E$ with coefficients in $\mathbb{Z}[q, q^n]$, given by $A_{-3}(E,Q,q) = \sum_{i=0}^6 B_i(q^n, q) E^i$.
• For the twist knot with $p = 3$, the non-commutative $A$-polynomial is a degree-6 $q$-difference operator with coefficients involving $q^{3n+15}$ and products of terms like $(q^{2n+1}-1)$, $(q^{2n+3}-1)$, etc.
• The method successfully computes the non-commutative $A$-polynomial for twist knots with $-8$ to $11$ crossings, which were previously inaccessible via standard methods.
• The computed non-commutative $A$-polynomials are monic, linear, and of minimal order, confirming their role as the characteristic polynomial of the $q$-holonomic colored Jones sequence.
• The specialization of the non-commutative $A$-polynomial at $q=1$ matches the known $A$-polynomial for twist knots, providing new evidence for the AJ-Conjecture.
• The method produces results that are both computationally manageable and structurally minimal, unlike traditional elimination techniques which fail for knots with more than 12 crossings.