Virtual knot groups and almost classical knots

Hans U. Boden, Anne Isabel Gaudreau, Eric Harper, Andrew J. Nicas, Lindsay White

Published 2015-06-04Version 1

We define a group-valued invariant of virtual knots and relate it to various other group-valued invariants of virtual knots. In particular, we show that it is isomorphic to both the extended group of Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. Given an almost classical knot, we show that the group factors as a free product of the usual knot group and Z. We establish a similar formula for mod p almost classical knots, and we use these results to derive obstructions to a virtual knot K being mod p almost classical. Virtual knots can be viewed as knots in thickened surfaces, and almost classical knots correspond to knots with homologically trivial representatives. Given an almost classical knot K, we construct a Seifert surface for K and use it to extend various classical knot invariants to the setting of almost classical knots. For instance, by considering the infinite cyclic cover of the complement of K, we show that the Alexander ideal is principal, a fact that was first proved by Nakamura, Nakanishi, Satoh, and Tomiyama using different methods. This gives a natural way to extend the Alexander polynomial to almost classical knots, and we prove that the resulting polynomial satisfies a skein relation and gives a bound on the Seifert genus of K. For mod p almost classical knots, the Alexander ideal can be shown to be principal over the ring Z[z_p], where z_p is a primitive p-th root of unity, and this allows us to define an analogue to the Alexander polynomial for these knots. We use parity to show that if K is mod p almost classical, then any minimal crossing diagram for it is mod p Alexander numberable. As an application, we tabulate almost classical knots up to 6 crossings and determine their Alexander polynomials and virtual genus.