Journal of Knot Theory and Its Ramifications

We show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. The construction embodied in the proof is applied to derive explicit formulas for the HOMFLY polynomials of some iterative families, including the torus links {T_n,3}_n≥0.

Kauffman's state summation formula is especially useful for computing the bracket polynomials of projection diagrams which are related by smoothings or crossing changes. This facilitates the writing of a symbolic algebra program which computes the normalized bracket polynomials and frequencies of knots and links whose projection diagrams result from a given knot's oriented projection diagram either by crossing changes or by orientation preserving smoothings called natural smoothings. These frequencies provide insight into the unknotting game (and similar resultant games) whose object is to specify crossing changes or natural smoothings that will transform a given projection diagram of a knot into a projection diagram representing an unknot (or some other specified knot or link). The practical utility of the state summation formula is greatly enhanced by means of diagrams for closed tangle sums. These diagrams offer a special cost-reducing method to obtain crucial information needed to compute the state summation formula. This special method also gives insight into why the bracket is unchanged by mutation and contributes a strategy to the enigmatic search for a non-trivial knot with Jones polynomial equal to one.

We prove the existence of a rationalisation of a classical or high-dimensional knot group Π which admits an involution if the Alexander polynomials of the knot are reciprocal. Using the group and its involution, we study the local structure, in the neighbourhood of an abelian representation, of the space of representation of the knot group Π in a a Lie group. We apply these results to the groups of classical prime knots up to 10 crossings.

For any n≥2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. In particular, our links have tunnel number n-1. We can also give infinitely many 2-component hyperbolic links with such properties.

In this paper we aim to set forth a notion of equivalence for 1-manifolds that are properly embedded in the space R×D² and prove that this is the same as ambient isotopy.

A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We show that irreducibility is preserved under certain deformations of embedded graphs. We show that certain embedded graphs are irreducible.