Stavros Garoufalidis
Abstract: Our paper originated from a generalization of the Volume Conjecture to multisums of q-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006. We introduce the notion of a special and general q-hypergeometric term (in short, special q-term and general q-term). The latter is a product of q-binomials and q-factorials in linear forms in several variables. In the first part of the paper, using elementary manipulations of symbols, we show how to assign elements of the Bloch group to a general q-term. The image of these elements under the Bloch-Wigner regulator map is always a finite subset of the set of purely imaginary periods, in the sense of Kontsevich-Zagier. In the second part of the paper, we extend our results to the extended Bloch group. The latter captures exactly the torsion information of $K_3^ind(C)$, and its regulator is given by the Rogers dilogarithm function. Our outcome is again a finite subset of the set of periods. In the third part of the paper, given a special q-term, we can associate two power series, convergent in a neighborhood of zero. We conjecture that the series have analytic continuation as a multivalued function in the complex numbers minus the above described finite set of points. Our conjecture implies a strong form of the Volume Conjecture (as shown in joint work with T.T.Q. Le) and is known to be true in case of 1-dimensional special q-terms, as is shown from joint work with O. Costin. In the final part of the paper, we describe a rich source of special terms that come from real 3-dimensional knotted objects. Finally we compare our combinatorial encodings of general q-terms with that of Neumann-Zagier and Kontsevich.
Key words: Bloch group, extended Bloch group, algebraic K-theory, regulators, dilogarithm, Rogers dilogarithm, Bloch-Wigner dilogarithm, potential, Bethe ansatz, special q-hypergeometric terms, asymptotic expansions, knots, homology spheres, Habiro ring, Volume Conjecture, periods, Quantum Topology, resurgence, Gevrey series.
Notes: 25 pages, 0 figures.