I will talk about some joint work with Andrew Neitzke, where we introduce a new "invariant" (with possible wall-crossing) for framed links in a three-manifold M=C \times R with C being an oriented surface. This invariant, denoted as F(L) for a framed link L, is valued in the GL(1) skein algebra of another three-manifold M'=C' \times R, where C' is an N-fold cover of C.
Under various special limits, F(L) turns into more familiar objects. When L is contained in a 3-ball in M, F(L) reproduces certain one-variable limit of the HOMFLY polynomial of L. When the projection of L to C has no crossings and the homology class of L is nontrivial, F(L) becomes a generating function encoding the spectrum of framed BPS states associated with certain half-BPS line defect in a 4d N=2 supersymmetric theory. In general, F(L) is a "hybrid" of the above two quantities.
The construction of F(L) is realized via a homomorphism from the GL(N) skein algebra of M to the GL(1) skein algebra of M'. In my talk I will first review the notion of skein algebras. Then I will describe this homomorphism for the special case of N=2, followed by some examples.