The Kashaev invariant, Nahm sums and modularity

The Kashaev invariant of a knot can be extended to a complex valued function on the set of complex roots of unity on the unit circle. Nahm sums are special $q$-hypergeometric sums defined inside the unit circle. Both are conjectured to have modular properties, and both properties are linked to an explicit map from the Bloch group, that we will discuss.
Joint work with Frank Calegar and Don Zagier. The author is supported in part by the National Science Foundation DMS-1406419.