Charles Frohman

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

The A-polynomial from the noncommutative viewpoint

The paper introduces a noncommutative generalization of the A-polynomial of a knot. This is done using the Kauffman bracket skein module of the knot complement, and is based on the relationship between skein modules and character varieties. The construction is possible because the Kauffman bracket skein algebra of the cylinder over the torus is a subalgebra of the noncommutative torus. The generalized version of the A-polynomial, called the noncommutative A-ideal, consists of a finitely generated ideal of polynomials in the quantum plane. Some properties of the noncommutative A-ideal and its relationships with the A-polynomial and the Jones polynomial are discussed. The paper concludes with the description of the examples of the unknot, and the right- and left-handed trefoil knots.

UNDERSTANDING THE KAUFFMAN BRACKET SKEIN MODULE

The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = eh/4. For a compact oriented surface F, it is shown that K(F×I) is a quantization of the -characters of the fundamental group of F corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined and obtained from topologically free Kauffman bracket modules. A structure theorem for K(M) is given in terms of the affine -characters of π1(M). It follows for compact M that K(M) can be generated as a module by cables on a finite set of knots. Moreover, if M contains no incompressible surfaces, the module is topologically finitely generated.

The ordering theorem for the ends of properly embedded minimal surfaces

A fundamental problem in the classical theory of minimal surfaces is to describe the asymptotic geometry of properly embedded minimal surfaces in . In the special case that the surface has finite total curvature its asymptotic behavior is well understood. For, in this case, the surface is conformally diffeomorphic to a finitely punctured closed Riemann surface and each end of the surface, one for each puncture point, is asymptotic to a plane or an end of a catenoid (see [19]). Thus the plane and the catenoid are the models for describing the asymptotic behavior of these minimal surfaces. When the properly embedded minimal surface has infinite total curvature, but still finite topology, the question has been asked whether the surface must be asymptotic to a helicoid. A first step towards understanding the asymptotic behavior of a surface is to characterize its topological behavior. For example, doubly and triply-periodic minimal surfaces in 3 that are not flat must have infinite genus and one end [2]. In [7] the authors’ proved that any two properly embedded minimal surfaces in 3 with the This research was supported by the National Science Foundation grant DMS-8701736. The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8900285. A surface has finite total curvature if ( ∫ |K|dA < ∞).

An intersection homology invariant for knots in a rational homology 3-sphere

THE purpose of this paper is to define a family of computable homological invariants of knots that generalize Casson’s invariant of knots. Let K be a homologically trivia1 knot in a rational homology 3-sphere N. Given a pair of integers (n, d) with n 2 1, we define a numerical invariant & and a related polynomial invariant pJt) which depend only on (N, K, n, d mod n). The invariant An,* can be thought of as an algebraic count of the number of characters of representations of the fundamental group of the complement of K into the Lie group SU(n) which take a longitude to eznid’” times the identity. The case where n and d are not relatively prime is of most interest to us here as the relatively prime case (for fibered knots) has been treated in [4]. While we define the invariants d see Theorems 5.21 and 5.22. Furthermore, an algorithm is given for determining these polynomials. We explicitly evaluate 1,. 0; see Theorem 6.4. For fibered knots, A,,, can be computed from the intersection homology Lefschetz number of the monodromy action on the moduli space of semistable holomorphic bundles of rank n and degree d and fixed determinant over a compact Riemann surface. For n and d not relatively prime, this moduli space is typically singular. Our computation relies heavily on the theory developed by Frances Kirwan ([ 14, 15, 16, 171) for desingularizing these spaces and computing their (mid-perversity) intersection homology. The polynomial invariants which we define in

Unitary representations of knot groups

3 are best understood in the abstract framework of “cobordism functors” developed in

A NOTE ON THE BAR-NATAN SKEIN MODULE

1. The axioms for such functors are somewhat reminiscent of, albeit less restrictive than, the axioms proposed for so-called “topological quantum field theories” (see Cl]). In [6] it was shown how the Alexander polynomial arises in an elementary fashion from U(1) representations in the context of cobordism functors (see [6, Theorem 4.41). Here, this is generalized to PU(n) representations from which we can obtain polynomial invariants which, at least in the case of fibered knots, are computable in terms of data derived from the Alexander polynomial. In order to put our theory into perspective, we first review the definition of Casson’s invariant. Let M be an oriented homology 3-sphere. Let H1 and Hz be two handlebodies so

TOPOLOGICAL INTERPRETATIONS OF LATTICE GAUGE FIELD THEORY

IN 1984 Casson introduced a new invariant for oriented homology three-spheres M. The invariant 1(M) is the algebraic number of irreducible W(2) representations of xi(M). The invariant reduces mod 2 to the p invariant. In the process of giving a method for computing his invariant he defined an invariant X(K) of oriented knots K in oriented homology spheres that is the algebraic number of PU(2) representations with nontrivial second Stiefel-Whitney class of the fundamental group of the result of longitudinal surgery on the knot. Casson’s invariant for knots reduces mod 2 to the arf invariant. I became interested in whether it was possible to find meaningful generalizations of Casson’s invariant by using Lie groups other than SU(2). The first step in such a program is to develop an analog of Casson’s invariant of knots. For each integer n 2 2 and each integer k there is an invariant of knots n(n, k) that gives an algebraic measure of the number of representations of the fundamental group of the result of longitudinal surgery on the knot into PU(n) so that the induced bundle on the manifold has first chern number congruent to k mod n. The definition of these invariants and the derivation of their properties is given in [4] and [S]. The material in this paper is the starting point for the computations there. In [4] it is shown that if some n(n, k) invariant is nonzero, where n and k are relatively prime then there exists an n 1 dimensional family of conjugacy classes of SU(n) representations of the fundamental group of the knot. In this paper I give a method of computing the n(n, k) invariants when K is a fibered knot and n and k are relatively prime. In this case n(n, k) is equal to a Lefschetz fixed point number defined in the body of the paper. The method of computation shows that the n(n, k) invariants of a fibered knot are determined by its Alexander polynomial. Furthermore the value of the Alexander polynomial at t = 1 and the n(n, k) invariants in turn determine the Alexander polynomial of a fibered knot in a rational homology sphere. The most striking result is Theorem 1.7. This theorem implies that if K is a fibered knot of genus g in a rational homology sphere then there exists an irreducible representation of the fundamental group of the knot complement into some SU(n) where 2 I n I g + 1. It is not unreasonable to conjecture that if K is a knot in a rational homology sphere and the Alexander polynomial of K is nontrivial, then a result similar to the one proved here for fibered knots should be true. Our method of computation is gauge theoretic in nature. It rests on the work of Atiyah and Bott [2] on Yang-Mills equations over a Riemann surface. The computation takes place in a direct summand of the cohomology of a classifying space for the gauge group of a hermitian bundle over a closed surface. Amazingly once the answer

A quantum obstruction to embedding

We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by Bar-Natan in [3], and modified by Khovanov [6]. We compute the structure of the modules for some manifolds, including Seifert fibered manifolds.

The Kauffman bracket skein as an algebra of observables

Abstract:We construct lattice gauge field theory based on a quantum group on a lattice of dimension one. Innovations include a coalgebra structure on the connections and an investigation of connections that are not distinguishable by observables. We prove that when the quantum group is a deformation of a connected algebraic group G (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of