20 research outputs found
Virtual knot groups and almost classical knots
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