729sl(2) tangle homology with a parameter and singular cobordisms CARMEN LIVIA CAPRAUWe construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl.2/ link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl.3/ theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl.2/ invariant (or Lee's modification of it) corresponds to a D 0 (or a D 1/: In particular, the construction naturally resolves the sign ambiguity in the functoriality of Khovanov's sl.2/ theory.

Abstract. Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander-and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.

Abstract. We show how to use Bar-Natan's 'divide and conquer' approach to computation to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.

We show that the virtual singular braid monoid on n strands embeds in a group V SGn, which we call the virtual singular braid group on n strands. The group V SGn contains a normal subgroup V SP Gn of virtual singular pure braids. We show that V SGn is a semi-direct product of V SP Gn and the symmetric group Sn. We provide a presentation for V SP Gn via generators and relations. We also represent V SP Gn as a semi-direct product of n − 1 subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.

We investigate the filtered theory corresponding to the universal sl(2) foam cohomology H * a,h for links, where a, h ∈ C. We show that there is a spectral sequence converging to H * a,h which is invariant under the Reidemeister moves, and whose E 1 term is isomorphic to Khovanov homology. This spectral sequence can be used to obtain from the foam perspective an analogue of the Rasmussen invariant and a lower bound for the slice genus of a knot.