Virtual knot groups and almost classical knots

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.Comment: 44 page

Bridge numbers for virtual and welded knots

“…(That E 0 = (0) follows from the fundamental identity and the fact that the presentation matrix A is square, see [6, (2.3)].) Notice that the length of the chain (3) of elementary ideals is bounded by the size of the Alexander matrix (2). For instance, if G K has meridional rank k, then using a presentation (1) of G K with k generators, then the associated Alexander matrix A is a k × k matrix, and it follows that E k = (1).…”

“…In this section, we show how to derive bounds on the virtual bridge number from the reduced virtual knot group G K . This is a group-valued invariant of virtual knots introduced in [2]. The group G K is isomorphic to two other the group valued invariants of virtual knots, namely the extended group π K of Silver and Williams [18] and the quandle group QG K introduced by Manturov in [11,Definition 3].…”

“…The group G K is isomorphic to two other the group valued invariants of virtual knots, namely the extended group π K of Silver and Williams [18] and the quandle group QG K introduced by Manturov in [11,Definition 3]. For details on the proof of this result, we refer to [2,Theorem 3.3]. Note that QG K coincides with the extended group of Bardakov and Bellingeri, cf.…”

“…The group G D is invariant under virtual isotopy, and we refer to [2] for the proof. The resulting invariant is called the reduced virtual knot group of K and will be denoted G K .…”

“…where A is the n × n Alexander matrix given in (2) and ∂r ∂v is the column vector with i-th entry equal to ∂r i ∂v a 1 ,...,an=t . The k-th elementary ideal is the ideal generated by all (n − k) × (n − k) minors of M , and they are invariants of the underlying module and form a nested sequence…”