First TQFT Mini-Workshop — 2014

Knot theory and number theory around the \(A\)-polynomial

The \(A\)-polynomials of knots were introduced in 1994 by Cooper-Culler-Gillet-Long-Shalen as a continuation of the inspirational work of Culler and Shalen which promoted the philosophy that the topological properties of a \(3\)-manifold \(M\) are reflected in the algebro-geometrical properties of the variety \(V\) of characters of the fundamental group of \(M\) into \(SL(2,\mathbb{C})\). Intuitively speaking, the \(A\)-polynomial of a knot \(K\) describes the variety \(V\) as viewed from the boundary of the \(3\)-manifold \(M\) obtained from the complement of \(K\) in the \(3\)-sphere \(S^3\).

Since 1994, the \(A\)-polynomial has proven to be a very useful invariant and has attracted a lot of researchers.

In this mini-workshop, we will focus on the connections between the \(A\)-polynomials and number theory (such as Mahler measures of \(A\)-polynomials, algebraic K-theory and dilogarithm) and the non-commutative or quantized generalization of the \(A\)-polynomial due to Garoufalidis (based on character varieties) and Gukov (based on quantum field theory), with its conjectured relation with the coloured Jones polynomials (AJ conjecture or quantum volume conjecture).

The mini-workshop will consist of a mini-course by M. H. Sengun (Warwick), supplemented by seminars.

Venue

Room 3.10, Mathematics Department, Instituto Superior Técnico, Lisbon.

Organizers

Nuno Freitas (Bayreuth), Roger Picken (IST, Lisbon), Mehmet Haluk Sengun (Warwick).

Support

Fundação para a Ciência e a Tecnologia (FCT), Centro de Análise Matemática, Geometria e Sistemas Dinâmicos (CAMGSD), Geometry and Mathematical Physics Project.