2011/01/31 by Stavros Garoufalidis, Christoph Koutschan
Mathematics · #Advanced Combinatorial Mathematics #Alexander polynomial #Algebraic structures and combinatorial models #Computation #Conjecture #Finite type invariant #Geometric and Algebraic Topology #Invariant (physics) #Jones polynomial #Knot invariant #Knot theory #math.CO #math.GT #msc:57M25 #msc:57N10
paper · pdf · doi:10.1080/10586458.2012.651409
published as Experimental Mathematics 21(3), pp. 241-251, 2012 · 12 pages, 8 figures. ISSN 1058-6458
openalex publication_date 2012/09/01 · arxiv created 2012/09/11 · arxiv updated 2012/09/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (noncommutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the Kp =(−2, 3, 3+2p) pretzel knots for p=−5, … , 5. This is a particularly interesting family, since the pairs (Kp , −K −p ) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the noncommutative A-polynomial complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, supports the AJ conjecture for knots with reducible A-polynomial, and numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariant to numerically verify the volume conjecture for the above-mentioned pretzel knots.