On the relation between the A-polynomial and the Jones polynomial

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.

     

Skein modules and the noncommutative torus

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 -th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.

     

加厚环面中标架环链的约化多项式

研究加厚环面中的标架环链.给出标架环链在Kauffman尖括号拆接代数中的表达式.利用Gro\"{o}bner基理论,我们从上述表达式中得到标架环链的约化多项式,该多项式是标架环链的同痕不变量且可计算.     

On Rational Knots and Links in the Solid Torus

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.     

The colored Jones polynomial and the A-polynomial of Knots

We study relationships between the colored Jones polynomial and the A-polynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial is established for a large class of two-bridge knots, including all twist knots. We formulate a weaker conjecture and prove that it holds for all two-bridge knots. Along the way we also calculate the Kauffman bracket skein module of the complements of two-bridge knots. Some properties of the colored Jones polynomial are established.     

The Kauffman bracket skein as an algebra of observables

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory.

     

On the Kauffman skein modules

 Let k be a subring of the field of rational functions in α, s which contains α ^±1 ,s ^±1 . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the k-module freely generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of , the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S ¹ × S ² is freely generated by the empty link, i.e., . Received: 20 October 2001 / Revised version: 20 March 2002     

Kauffman bracket skein module of a connected sum of 3-manifolds

We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds. Received: 12 January 1998 / Revised version: 15 September 1999     

A finite set of generators for the Kauffman bracket skein algebra

If F is a compact orientable surface it is known that the Kauffman bracket skein module of has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes . Received November 27, 1995; in final form September 29, 1997     

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Let M be a compact orientable 3-manifold. The set of characters of SL ₂()-representations of forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module, modulo its nilradical. This is accomplished by realizing the module as a combinatorial analog of the ring in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional. Received: April 18, 1996     

Traces on the skein algebra of the torus

For a surface F, the Kauffman bracket skein module of F×[0,1], denoted K(F), admits a natural multiplication which makes it an algebra. When specialized at a complex number t, nonzero and not a root of unity, we have Kt(F), a vector space over C. In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space Kt(T²) has five distinct traces. One trace, the Yang-Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on Kt(T²) correspond to the four singular points of the moduli space of flat SU(2)-connections on the torus.     

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Multiplicative structure of Kauffman bracket skein module quantizations

We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric algebra in three variables (and an algebra closely related to a cyclic quantization of )). For a torus without boundary we obtain a quantization of ``the symmetric homologies" of a torus (equivalently, the coordinate ring of the -character variety of ). Presentations are also given for the four-punctured sphere and twice-punctured torus. We conclude with an investigation of central elements and zero divisors.

     

Tabulation of Prime Knots in Lens Spaces

Using computational techniques, we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces \(L(p,q)\). For these knots, we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for \(p\ge 13\).     

The Mod 2 Kauffman Bracket Skein Module of Thickened Torus

Framed links in thickened torus are studied. We define the mod 2 Kauff-man bracket skein module of thickened torus and give an expression of a framed link in this module. From this expression we propose a new ambient isotopic invariant of framed links.     

The Yang-Mills measure in the Kauffman bracket skein module

For each closed, orientable surface , we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module . The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = − 1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of . Received: June 2, 2000     

An invariant of links in the thickened torus

An invariant of links with two and more components in the thickened torus is constructed; the invariant depends on several variables. The construction uses Kauffman’s formal theory, which is based on Dehn’s representation of knot groups. This invariant is a natural generalization of a polynomial z constructed by Zenkina and Manturov. Some properties of the new invariant are also considered.     

On the calculation of the Kauffman bracket polynomial

In this paper, we present a new algorithm to evaluate the Kauffman bracket polynomial. The algorithm uses cyclic permutations to count the number of states obtained by the application of ‘A’ and ‘B’ type smoothings to the each crossing of the knot. We show that our algorithm can be implemented easily by computer programming.     

Estimating a skein module with characters

We introduce a new technique for estimating the number of generators of the Kauffman bracket skein module of a three manifold; one which requires the construction of linear functionals on a simpler version of the module. Of particular interest is the use of representations of the fundamental group into to generate the functionals.

     

Invariants of B-Type Links Via an Extension of the Kauffman Bracket

We construct invariants of solid torus links ( -type invariants) generalizing the Kauffman bracket. These invariants give us expressions of statistical sums in some special cases. Bibliography: 29 titles.