2007/07/31 by Garoufalidis, Stavros
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.0708.0018
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; \citeKo. We introduce the notion of a q-hypergeometric term (in short, q-term). The latter is a product of ratios of q-factorials in linear forms in several variables. In the first part of the paper, we show how to construct elements of the Bloch group (and its extended version) given a \qterm. Their image under the Bloch-Wigner map or the Rogers dilogarithm is a finite set of periods of weight 2, in the sense of Kontsevich-Zagier. In the second part of the paper we introduce the notion of a special q-term, its corresponding sequence of polynomials, and its generating series. Examples of special q-terms come naturally from Quantum Topology, and in particular from planar projections of knots. The two parts are tied together by a conjecture that relates the singularities of the generating series of a special q-term with the periods of the corresponding elements of the extended Bloch group. In some cases (such as the 41 knot), the conjecture is known.