Virtual knot groups and almost classical knots

TL;DR

This paper introduces a new group-valued invariant for virtual knots, explores the class of almost classical knots, and establishes their properties, including Alexander invariants, Seifert surfaces, and genus bounds, with classifications up to 6 crossings.

Contribution

It defines a new invariant for virtual knots, characterizes almost classical knots, and derives their algebraic and geometric properties, including Alexander invariants and genus bounds.

Findings

The group factors as a free product of the usual knot group and Z for almost classical knots.

The Alexander ideal for almost classical knots is principal, and the Alexander polynomial satisfies a skein relation.

Classification of almost classical knots up to 6 crossings with computed invariants.

Abstract

We define a group-valued invariant of virtual knots and relate it to various other group-valued invariants of virtual knots, including the extended group of Silver-Williams and the quandle group 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 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. 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 6 crossings and determine their Alexander polynomials and virtual genus.