The Khovanov space and generalizations

A quantum knot cohomology is a knot invariant recovering a quantum knot polynomial as its Euler characteristic.  Sometimes these cohomologies are the usual singular cohomologies of spaces which are themselves knot invariants.  The first example is due to Lipshitz and Sarkar: the Khovanov space.  I'll tell you why you might care about this if you're only interested in low-dimensional topology.  I'll also sketch the construction, aiming to keep it understandable, and point to some generalizations.  No knowledge assumed.  This is joint work in various combinations with Feller, Jones, Lewark, Orson, and Schuetz.