Ramanujan, K-theory, and modularity

The Rogers-Ramanujan identity:

$$ 1 + \frac{q}{(1-q)} + \frac{q^4}{(1-q)(1-q^2)} + \frac{q^9}{(1-q)(1-q^2)(1-q^3)} + \ldots =
\frac{1}{(1-q)(1-q^4)(1-q^6)(1 - q^9) \ldots}$$

says that a certain $q$-hypergeometric function (the left hand side) is equal to a modular form (the right hand side). To what extent can one classify
all $q$-hypergeometric functions which are modular? We discuss this question and its relation to conjectures in knot theory and K-theory.
This is joint work with Stavros Garoufalidis and Don Zagier.