Stavros Garoufalidis and Xinyu Sun
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 $(J(n))$ and the hypergeometric kernel $c(n,k)$. General theory predicts the existence of a computable recursion relation for $J(n)$; however not one of minimal order in general. As an application of our method, we 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.
Key words: Knots, Jones polynomial, colored Jones function, $A$-polynomial, $C$-polynomial, non-commutative $A$-polynomial, $q$-difference equations, WZ Algorithm, Creative Telescoping, Gosper's Algorithm, certificate, multi-certificate.
Notes: 18 pages with 1 figure.