Abstract. We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the n-th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.
The non-commutative generalization of the A-polynomial of a knot of Cooper,
Culler, Gillet, Long and Shalen [4] was introduced in [6]. This generalization consists
of a finitely generated left ideal of polynomials in the quantum plane, the non-
commutative A-ideal, and was defined based on Kauffman bracket skein modules, by
deforming the ideal generated by the A-polynomial with respect to a parameter. The
deformation was possible because of the relationship between the skein module with
the variable t of the Kauffman bracket evaluated at −1 and the SL(2, C)-character
variety of the fundamental group, which was explained in [2]. The purpose of the
present paper is to compute the non-commutative A-ideal for the left- and right-
handed trefoil knots. As will be seen below, this reduces to trigonometric operations in
the non-commutative torus, the main device used being the product-to-sum formula
for non-commutative cosines.
Abstract. Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the theory of classical theta functions. It turns out that the theory of theta functions, from the representation theoretic point of view of A. Weil, is just an instance of Chern-Simons theory. The group algebra of the finite Heisenberg group is described as an algebra of curves on a surface, and its Schrödinger representation is obtained as an action on curves in a handlebody. A careful analysis of the discrete Fourier transform yields the Murakami-Ohtsuki-Okada formula for invariants of 3-dimensional manifolds. In this context, we give an explanation of why the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds under gluings obey the same formula.
Abstract. This paper shows that the noncommutative generalization of the A-polynomial of a knot, defined using Kauffman bracket skein modules, together with finitely many colored Jones polynomials, determines the remaining colored Jones polynomials of the knot. It also shows that under certain conditions, satisfied for example by the unknot and the trefoil knot, the noncommutative generalization of the A-polynomial determines all colored Jones polynomials of the knot.
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