Narrow Results

We show that the values at t = 1 of the first r coefficient polynomials of the HOMFLY polynomial of an r-component link L are determined by the linking numbers. This generalizes a formula of Hoste and one of Lickorish and Millett. The formula was proved before, using a different method, by J. Przytycki and related to homotopy skein modules [15].

In these lectures we present for the first time a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, and we provide also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction offered here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism and it is complemented by the independent analysis and discussion made by Maxwell in 1867. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important rôle in modern mathematical physics, we offer a direct proof of their equivalence, and we provide some examples of linking number computation based on oriented area information. The material presented in these lectures forms an integral part of a paper that will appear in the Journal of Knot Theory and Its Ramifications.

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial.