q-Terms, singularities and the extended Bloch group

TL;DR

This paper generalizes the Volume Conjecture to multisums of q-hypergeometric terms, constructing elements of the extended Bloch group and relating their singularities to periods, with applications in Quantum Topology.

Contribution

It introduces the notion of q-terms and connects their properties to the extended Bloch group and quantum topology, providing new insights into their structure and singularities.

Findings

Constructed elements of the extended Bloch group from q-terms.

Linked singularities of generating series to periods of Bloch group elements.

Validated the conjecture for specific cases like the 4_1 knot.

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; \cite{Ko}. 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 $4_1$ knot), the conjecture is known.