TheC–polynomial of a knot

Abstract

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q -difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A-polynomial of a knot.In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group U q .sl 2 /) specializes at q D 1 to the better known A-polynomial of a knot, which has to do with genuine SL 2 ‫/ރ.‬ representations of the knot complement.Computing the non-commutative A-polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the C -polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative A-polynomial of twist knots. Finally, we formulate a number of conjectures relating the A, the C -polynomial and the Alexander polynomial, all confirmed for the class of twist knots. 57N10; 57M25

Super-A-polynomials for twist knots

“…7 The algorithm which computes difference equations of minimal order for multi-sum expressions has been developed by S. Garoufalidis and X. Sun [67,72,73]. By the algorithm with a certicate, the quantum A-polynomials were obtained for the twist knots A(K p ;x,ŷ; q) in the range −14 ≤ p ≤ 15 [67].…”

The colored Jones function is q-holonomic

“…For computations of recursion relations of the cyclotomic function of twist knots, we refer the reader to [12].…”

Irreducibility ofq–difference operators and the knot 74

“…With our definition, we have c −1,n (q) = 1 and c 1,n (q) = (−1) n q n(n+3)/2 . In [GS06] it was shown that for each integer p, the sequence c p,n (q) satisfies a monic recurrence of order |p| with initial conditions c p,n (q) = 0 for n < 0 and c p,0 (q) = 1. In particular, for p = 2 we have: We employ the definition of J K p,p ′ ,n (q) given in (3) and (4) in terms of definite sums to compute a recurrence for the colored Jones polynomial of 7 4 = K 2,2 .…”