Narrow Results

A rack is a non-empty set with a binary operation such that right multiplication is an automorphism. New racks closely related to the braid groups and the mapping class groups will be introduced using "cords" on surfaces, and their structure will be investigated. Emphasis will be laid on the racks of cords on the plane and the sphere. In particular, their presentations and the center of the associated group of the rack of cords on the sphere will be given.

Knot and link diagrams are used to represent nonstandard sets, and to represent the formalism of combinatory logic (lambda calculus). These diagrammatics create a two-way street between the topology of knots and links in three dimensional space and key considerations in the foundations of mathematics.

A traditional method of finding topological invariants of a knot in 3-dimensional euclidean space is to colour the arcs of a planar diagram and use this to extract invariants of the knot such as the fundamental group. This method can be extended in many ways to other kinds of codimension 2 embeddings. Moreover the new algebraic objects which arise quite naturally from these generalisations such as quandles, racks, biquandles etc are of interest in their own right. In this paper we shall give a brief overview of the methods and the new algebraic structures.