The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful as those polynomials. As invariants of finite type are much easier to define and manipulate than the quantum group invariants, it is likely that in attempting to classify knots, invariants of finite type will play a more fundamental role than the various knot polynomials.

Abstract. We study the perturbation theory for three dimensional Chern–Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the 2-loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the 1-loop case [28]. In fact, the counterterm is equal to the Chern–Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten’s exact solution.

For a copy with the hand-drawn figures please email

. We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander-Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin-Morton-Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Contents 1. Introduction 2 1.1. The conjecture 1.2. Preliminaries 1....

To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is first-order acyclic sharing graphs represented by let-syntax, and others are extensions with higher-order constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced

Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig’s canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger. 1.

. Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi / Power and Robinson and traced monoidal categories by Joyal, Street and Verity. The former is used for representing the notion of sharing, whereas the latter for cyclic data structures. Our new models provide a semantic framework for understanding recursion created from cyclic sharing, which includes traditional models for recursion created from fixed points as special cases. Our cyclic lambda calculus serves as a uniform language for this wider range of models of recursive computation. 1 Introduction One of the traditional methods of interpreting a recursive program in a semantic domain is to use the least fixed-point of continuous functions. However, in the real implementations of program...

In recent years, it has been discovered that invariants of three dimensional

In [2], [3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian Chern-Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the ‘free semistrict braided monoidal 2-category with duals on one unframed self-dual object’. By this universal property, any unframed self-dual object in a braided monoidal 2-category with duals determines an invariant of 2-tangles in 4 dimensions. 1