The Non-commutative A-polynomial of Twist Knots
نویسندگان
چکیده
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) = P k c(n, k)Ĵ(k) given a recursion relation for (Ĵ(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.
منابع مشابه
The skein module of torus knots complements
We compute the Kauffman skein module of the complement of torus knots in S3. Precisely, we show that these modules are isomorphic to the algebra of Sl(2, C)-characters tensored with the ring of Laurent polynomials. Skein modules were introduced indenpendantly by V. Turaev in 1988 and J. Przytycki in 1991 (see [TU88, HP92]) as a C[A±1]-module associated to a 3-manifold M generated by banded link...
متن کاملNon–abelian Reidemeister Torsion for Twist Knots
Twist knots form a family of special two–bridge knots which include the trefoil knot and the figure eight knot. The knot group of a two–bridge knot has a particularly nice presentation with only two generators and a single relation. One could find our interest in this family of knots in the following facts: first, twist knots except the trefoil knot are hyperbolic; and second, twist knots are n...
متن کاملar X iv : m at h / 06 06 25 5 v 1 [ m at h . G T ] 1 1 Ju n 20 06 COEFFICIENTS AND NON - TRIVIALITY OF THE JONES POLYNOMIAL
We show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. Then we prove that there are infinitely many positive knots with no positive minimal crossing diagrams, and infinitely many achiral knots of odd crossing number. Some applications to the twist number of a link, Mahler measure and the hyperbolic volu...
متن کاملQUATERNIONIC INVARIANTS of VIRTUAL KNOTS and LINKS
In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2×2 matrices with entries in a possibly non-commutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot. The contents of the paper are as follows
متن کاملA-polynomials of a Family of Two-bridge Knots
The J(k, l) knots, often called the double twist knots, are a subclass of two-bridge knots which contains the twist knots. We show that the A-polynomial of these knots can be determined by an explicit resultant. We present this resultant in two different ways. We determine a recursive definition for the A-polynomials of the J(4, 2n) and J(5, 2n) knots, and for the canonical component of the A-p...
متن کامل