David Joyce

A classifying invariant of knots, the knot quandle

Abstract The two operations of conjugation in a group, x▷y=y -1 xy and x▷ -1 y=yxy -1 satisfy certain identities. A set with two operations satisfying these identities is called a quandle. The Wirtinger presentation of the knot group involves only relations of the form y -1 xy = z and so may be construed as presenting a quandle rather than a group. This quandle, called the knot quandle, is not only an invariant of the knot, but in fact a classifying invariant of the knot.

Generalized chromatic polynomials

Abstract There are several polynomials used in the study of combinatorics including Birkhoffs chromatic polynomial for graphs, Stanleys order polynomial for partially ordered sets, and Tuttes dichromatic polynomial for graphs. We develop a common basis for these polynomials which depends on a sequence of standard reference objects in the category under consideration, the sequence being called a chromatic complex. The values of the polynomial then indicate the number of morphisms in the category from the object in question to the objects in the chromatic complex.

Mathematical concepts for analyzing ambiguous situations

Abstract We examine some mathematical tools for dealing with ambiguous situations. The main tool is the use of non-standard logic with truth-values in what is called a locale. This approach is related to fuzzy set theory, which we briefly discuss. We also consider probabilistic concepts. We include specific examples and describe the way a researcher can set up a suitable locale to analyse a concrete situation.