The original construction of the Khovanov homology of a link can be seen as a formal complex in the category of flat tangles and surfaces between them. There is a way to associate a chain map with a link cobordism, but only up to a sign. Blanchet has fixed this by introducing the category of gl(2)-foams, certain singular cobordisms between planar trivalent graphs. Originating from the representation theory of quantum groups, foams are usually thought as algebraic objects. In my talk I will bring topology back by interpreting foams as two surfaces transverse to each other. This description leads to a quick proof that gl(2)-foams can be evaluated, a construction of a natural basis of foams, and an explicit equivalence between the category of gl(2)-foams and cobordisms between flat tangles [1]. An immediate application is a functorial version of the Chen-Khovanov tangle homology as well as of the quantized annular homology, constructed previously by Anna Beliakova, Stephan Wehrli, and me [2].

This is a joint work with Anna Beliakova, Matthew Hogancamp and Stephan Wehrli.

References:
[1] arXiv:1903.12194
[2] arXiv:1605.03523