Virtual knot groups and almost classical knots

Hans U. Boden ¹   ; Robin Gaudreau ¹   ; Eric Harper ¹   ; Andrew J. Nicas ¹   ; Lindsay White ¹

¹ Mathematics & Statistics McMaster University Hamilton, Ontario L8S 4K1, Canada

Fundamenta Mathematicae, Tome 238 (2017) no. 2, pp. 101-142

Résumé

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.

Détail

DOI : 10.4064/fm80-9-2016

Keywords: define group valued invariant overline virtual knots overline mathbin * mathbb mathbb where denotes virtual knot group introduced boden further overline isomorphic extended group silver williams quandle group manturov bardakov bellingeri virtual knot called almost classical admits diagram alexander numbering that overline splits k* mathbb where knot group establish similar formula mod nbsp almost classical knots derive obstructions being mod nbsp almost classical viewed knots thickened surfaces almost classical knots correspond those homologically trivial admit seifert surfaces relate their alexander invariants homology associated infinite cyclic cover prove first alexander ideal principal recovering result first proved nakamura nbsp using different methods resulting alexander polynomial shown satisfy skein relation its degree gives lower bound seifert genus tabulate almost classical knots six crossings determine their alexander polynomials virtual genus