Khovanov homology and star-like isotopies (Q1009755)

In a previous paper [Algebr. Geom. Topol. 5, 1535--1553 (2005; Zbl 1084.57010)], the author and \textit{T. Fiedler} defined a refinement of the Jones polynomial and the Khovanov homology that is invariant under braid-like isotopy. These are the Reidemeister moves on a diagram of a knot that correspond locally to braid isotopies; in particular Reidemeister moves \(I\), \(IIb\) and Markov moves are not allowed. In this paper the author gives a second refinement \(V_{st}\) of the Jones polynomial and the Khovanov homology the for ``star-like'' isotopies. Here Markov moves are exluded, but pairs of Reidemeister moves \(I\) are allowed, star-like \(II\)-moves are \(II\)-moves corresponding to braid isotopies, and star-like \(III\)-moves are the Reidemeister \(III\)-moves that are not braid like. \(V_{st}\) is definded in terms of Kauffman states of a diagram.