Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots

Abstract

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knot and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25] [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial are nonclassical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov-Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen's result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in S g × I × I in the sense of Turaev [42].

Rotational virtual knots and quantum link invariants

“…This implies (by properties of the quandle) that virtual knots obtained in this way from classical non-trivial knots will themselves be non-trivial. In fact, such knots are nonclassical [5]. It is an open problem whether there are classical knots (actually knotted) having unit Jones polynomial.…”

“…Examples produced by virtualization are guaranteed to be non-trivial. It is more difficult to prove in general [5] or even in special cases that such virtualizations are non-classical. This has led to the discovery of new invariants for virtual knots.…”