Units, K-theory, and quantum invariants of knots

In recent years, there has been intense interest in the quantum invariants of knots and their asymptotic properties, a typical example being the celebrated Volume Conjecture for the Kashaev invariant. But it turns out there are also very interesting arithmetic properties of these invariants, including a surprising near-modular transformation property. Even though many of these are only conjectural, one can check them numerically to high precision, and when one does this, algebraic numbers of a special sort (roots of units in certain number fields) appear by magic. This led, in joint work with Frank Calegari and Stavros Garoufalidis, to a new (non-conjectural) construction of units starting from elements in so-called Bloch groups, and as a side product also to a solution of Nahm's conjecture on the modularity of certain special q-hypergeometric series.