The Khovanov Complex for Virtual Links

In the last few years, knot theory has enjoyed a rapidly developing generalisation, the Virtual knot theory, proposed by Louis Kauffman in 1996, see [Kau2]. A virtual link is a combinatorial generalisation of the notion of classical links: we consider planar diagrams with a new crossing type allowed; this new crossing (called virtual and marked by a circle) is neither an overcrossing nor an undercrossing. It should be treated as an artefact of two branches, which do not want to intersect but can not do without. This leads to a natural generalisation of Reidemeister moves for the virtual case: besides usual ones (which should be treated as local transformation inside a small 3-dimensional domain), we also add a detour move, which means the following. If there is arc of the diagram between some points A and B contains only virtual crossings, it can be removed and detoured as any other path connecting A and B; all crossings which occur in this path are set to be virtual as shown in Fig. 1. Thus, a virtual link is an equivalence class of virtual diagrams modulo generalised Reidemeister moves, the latter consisting of classical Reidemeister moves and the detour move.