Vassiliev theory and regional change
نویسنده
چکیده
The purpose of this note is to state some definitions that may be useful in the study of knots, manifolds and the like. They apply to anything for which the concept of a regional change can be defined, such as a product of elements in a group. The motivation comes from the Vassiliev theory for invariants of knots in R, the general nature of the axioms for a topological quantum field theory, and an observation regarding the Kontsevich knot integral. The basic concept is that of a regional change. To change a crossing in a knot diagram is an example of a regional change. This provides a choice. Assume that one option is labelled a (for above) and the other b (for below). Call such a double point. (This is not quite the standard usage.) Let k be a knot with r double points. The double points are assumed ordered, and labelled 1 to r. Now for each word w in a and b of length r let k be the actual knot formed by taking the wi choice at the i-th double point. Let K denote the group of formal sums of (isomorphism types of) knots in R3, with integer coefficients. Each knot k with r regional changes (or double points) determines a formal sum
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