Type Construction and Cycle Detection

This is a survey of results on the construction of holomorphic cusp forms on tube domains originally initiated by Ikeda. Besides a survey it includes conjectures and possible applications of our work.

We describe the first part of a gluing theory for the bigraded Khovanov homology with integer coefficients. This part associates a type D structure to a tangle properly embedded in a half-space and proves that the homotopy class of the type D structure is an invariant of the isotopy class of the tangle. The construction is modeled off bordered Heegaard-Floer homology, but uses only combinatorial/diagrammatic methods