Virtual knot groups and almost classical knots
Volume 238 / 2017
Abstract
We define a group-valued invariant $\overline G_K$ of virtual knots $K$ and show that $VG_K=\overline G_K \mathbin{*_{\mathbb Z}\,} \mathbb Z^2,$ where $VG_K$ denotes the virtual knot group introduced by Boden et al. We further show that $\overline G_K$ is isomorphic to both the extended group $EG_K$ of Silver–Williams and the quandle group $QG_K$ of Manturov and Bardakov–Bellingeri.
A virtual knot is called almost classical if it admits a diagram with an Alexander numbering, and in that case we show that $\overline G_K$ splits as $G_K*\mathbb Z$, where $G_K$ is the knot group. We establish a similar formula for mod $p$ almost classical knots and derive obstructions to $K$ being mod $p$ almost classical.
Viewed as knots in thickened surfaces, almost classical knots correspond to those that are homologically trivial. We show they admit Seifert surfaces and relate their Alexander invariants to the homology of the associated infinite cyclic cover. We prove the first Alexander ideal is principal, recovering a result first proved by Nakamura et al. using different methods. The resulting Alexander polynomial is shown to satisfy a skein relation, and its degree gives a lower bound for the Seifert genus. We tabulate almost classical knots up to six crossings and determine their Alexander polynomials and virtual genus.
