Quantum invariants of knotted surfaces

Following Yoshikawa we describe any knotted surface in four space by inserting saddle points into a single (usual) link diagram. In addition to the usual Reidemeister moves there are a few additional moves involving the saddles and from this perspective one can try extend any link invariant to an invariant of knotted surfaces. Given their local definition, quantum invariants seem especially suitable: the task is to find a suitable tensor representing the saddle and solving Yang-Baxter-like equations. We will show how to carry out this program in the case of Kashaev's metaplectic invariants using Gaussian integration. There are strong relations to the Alexander polynomial, the signature and the Rozansky-Overbay invariant. This is joint work in progress with Rinat Kashaev. A handout for this talk will be available at www.rolandvdv.nl/Talks/Banff26