Journal of Knot Theory and Its Ramifications

We tabulate all prime alternating knots through fourteen crossings. This provides the first list of 14-crossing prime alternating knots as well as the first confirmation of Thistlethwaite's tabulation of 12- and 13-crossing alternating knots.

The complete description of the Alexander polynomial of the complement of an irreducible sextic in is given. Some general results about Alexander polynomials of algebraic curves are also obtained.

A Gaussian random walk is a random walk in which each step is a vector whose coordinates are Gaussian random variables. In 3-space, if a Gaussian random walk of n steps begins and ends at the origin, then we can join successive points by straight line segments to get a knot. It is known that if n is large, then the knot is non-trivial with high probability. We give a new proof of this fact. Our proof shows in addition that with high probability the knot is contained as an essential loop in a fat, knotted, solid torus. Therefore the knot is a satellite knot and cannot be unknotted by any small perturbation.

We introduce a generalization of spin models by dropping the symmetry condition. The partition function of a generalized spin model on a connected oriented link diagram is invariant under Reidemeister moves of type II and III, giving an invariant for oriented links.

This paper describes the development and implementation of an algorithm which uses simulated annealing to recognize knots by minimizing an energy function defined over all knots. A knot is represented by a piecewise linear curve and the vertices of this curve are perturbed using simulated annealing to minimize the energy. Moving one line segment through another line segment is prohibited. The resulting minimum energy configuration is defined to be the canonical form. The algorithm is then tested with two different types of energy over a collection of complex knots.

A p. 1. proper knot is a proper p. 1. embedding of the real line into a p. 1. open 3-manifold. Two p. 1. proper knots are said to be equivalent if there is a p. 1. proper isotopy (possibly non-ambient) connecting them. A priori such an isotopy need not be locally flat: for example a local knot in the range could be shrunk to a point within a 3-cell. In this paper we show that, in essence, this is the only type of non-locally flat phenomena. We show that such a local knot can always be "combed to infinity". Thus if two p. 1. proper knots are connected by a p. 1. proper isotopy then the connecting isotopy can be modified to be a locally flat p. 1. proper isotopy. Hence, in proper knot theory, the smooth category and the p. 1. category are similar.

The relationship between Braid Theory and the organisation of periodic orbits of dynamical systems is considered.

It is shown that for some (physically relevant) 3-d flows the characterisation of periodic orbits by means of Braid Theory can be done on the Poincaré surface in an efficient way. The result is a thread-less graphical presentation of a braid class.

We discuss extensions of this approach to (adequate) dynamical systems of dimension higher than three, using results from Central Manifold Theory.

We study the relation between the two notions of index of graphs defined in [1]. We show that in the geometrically significant context of bipartite graphs the two coincide.

R. Bott defined two combinatorial invariants for finite cell complexes in the 1950's [2]. As described in [3], they fit beautifully into a state model framework. Recently, it has been observed that for finite graphs, the coefficients of this polynomial are the same set of invariants as defined by H. Whitney in [5]. In section 1, we define the Bott-Whitney polynomial and prove some of its basic properties. In section 2, we show that for a planar connected graph, the Bott-Whitney polynomial is essentially the chromatic polynomial of its dual graph. In section 3, an interpretation of the coefficients of the Bott-Whitney polynomial is given following Whitney [5]. In section 4, we study more general Bott polynomials.

The Reshetikhin-Turaev representation of the mapping class group of an orientable surface is computed explicitly in the case r = 4. It is then shown that the restriction of this representation to the Torelli group is equal to the sum of the Birman-Craggs homomorphisms. The proof makes use of an explicit correspondence between the basis vectors of the representation space, and the Z/2Z-quadratic forms on the first homology of the surface. This result corresponds to the fact, shown by Kirby and Melvin, that the three-manifold invariant when r = 4 is related to spin structures on the associated four-manifold.