2008/02/27 by Stavros Garoufalidis, Xinyu Sun, Garoufalidis, Stavros +1
Mathematics · #57M25 #57N10 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.CO #math.GT #msc:57M25 #msc:57N10
paper · pdf · doi:10.48550/arxiv.0802.4074
AMS-LaTeX, 18 pages with 1 figure
openalex publication_date 2008/02/27 · arxiv created 2009/07/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)=∑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.