Modelling supercoiled DNA knots and catenanes by means of a new regular isotopy invariant

This paper extends the knot polynomial classification of DNA knots and catenanes, by incorporating a measure of supercoiling. Cozzarelli, Millett, and White have used the Jones polynomial and the generalised 2-variable polynomial to describe the products of iterated Tn3 resolvase recombination and phage integrase mediated recombination. A new polynomial invariant, , is introduced; based on the regular isotopy invariants of Kauffman. The polynomial is an invariant of framed links and involves the Whitney degree of the link. This is useful because it not only allows a regular isotopy classification, but also distinguishes between plectonemic and solenoidal supercoils. The enzymes above require plectonemic supercoils for the synaptic substrate, so we put the polynomial to use to investigate the supercoiling of the recombination products.