Integral TQFT for a one-holed torus

We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expansion gives an approximation of the representation by representations into finite groups. As a special case, we study the induced representations over finite fields and identify them up to isomorphism. The key technical ingredient of the paper are new bases of the Integral TQFT modules which are orthogonal with respect to the Hopf pairing. We construct these orthogonal bases in arbitrary genus, and briefly describe some other applications of them.Comment: 18 pages, 8 figures. version 3: Minor expository changes. Bibliography update

Skein-theoretical derivation of some formulas of Habiro

We use skein theory to compute the coefficients of certain power series considered by Habiro in his theory of sl_2 invariants of integral homology 3-spheres. Habiro originally derived these formulas using the quantum group U_q sl_2. As an application, we give a formula for the colored Jones polynomial of twist knots, generalizing formulas of Habiro and Le for the trefoil and the figure eight knot.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-17.abs.htm

A New Matrix-Tree Theorem

The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably defined matrix. This result can be interpreted topologically as an expression for the lowest order term of the Alexander-Conway polynomial of an algebraically split link. We also prove some algebraic properties of our Pfaffian-tree polynomial.Comment: minor changes, 29 pages, version accepted for publication in Int. Math. Res. Notice

Maslov index, Lagrangians, Mapping Class Groups and TQFT

Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.Comment: 31 pages, 11 Figures. to appear in Forum Mathematicu

On the optimality of the Arf invariant formula for graph polynomials

We prove optimality of the Arf invariant formula for the generating function of even subgraphs, or, equivalently, the Ising partition function, of a graph.Comment: Advances in Mathematics, 201

Integral Lattices in TQFT

We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO(3) TQFT-modules of surfaces at roots of unity of odd prime order. Some applications relating quantum invariants to classical 3-manifold topology are given.Comment: 31 pages, v2: minor modifications. To appear in Ann. Sci. Ecole Norm. Su