A splitting formula for the spectral flow of the odd signature operator on 3-manifolds coupled to a path of connections.
A twisted dimer model for knots
We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.
A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds
In this paper we show that given any 3-manifold and any non-fibered class in there exists a representation such that the corresponding twisted Alexander polynomial is zero. We obtain this result by extending earlier work of ours and by combining this with recent results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.
A volume form on the SU(2)-representation space of knot groups.
Adequacy of Link Families
Alexander polynomial and Koszul resolution.
Alexander polynomial, finite type invariants and volume of hyperbolic knots.
Algebraic linking numbers of knots in 3-manifolds.
All roots of unity are detected by the A-polynomial.
Almost integral TQFTs from simple Lie algebras.
An almost-integral universal Vassiliev invariant of knots.
An infinite torus braid yields a categorified Jones-Wenzl projector
A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.
An introduction to the abelian Reidemeister torsion of three-dimensional manifolds
These notes accompany some lectures given at the autumn school “Tresses in Pau” in October 2009. The abelian Reidemeister torsion for -manifolds, and its refinements by Turaev, are introduced. Some applications, including relations between the Reidemeister torsion and other classical invariants, are surveyed.
An invariant of tangle cobordisms via subquotients of arc rings
We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
An operator invariant for handlebody-knots
A handlebody-knot is a handlebody embedded in the 3-sphere. We improve Luo's result about markings on a surface, and show that an IH-move is sufficient to investigate handlebody-knots with spatial trivalent graphs without cut-edges. We also give fundamental moves with a height function for handlebody-tangles, which helps us to define operator invariants for handlebody-knots. By using the fundamental moves, we give an operator invariant.
Analyticity of the free energy of a closed 3-manifold.
Asymptotic Vassiliev invariants for vector fields
We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of . More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field.
Automorphic forms and rational homology 3-spheres.