arXiv subjects
Publications and source records attributed to Tatsuro Shimizu.
The defect $d(M,ρ)$ is an invariant of a compact oriented 3-manifold $M$ with a representation $ρ$ of the fundamental group. In this article we give a diagrammatic method for $d$ of knot exteriors by using knot diagrams.
The Chern-Simons perturbation theory gives an invariant $d(M,ρ)$ for a pair of a closed oriented 3-manifold $M$ and a representation $ρ$ of the fundamental group. We generalize $d(M,ρ)$ for compact oriented 3-manifolds with torus boundaries. Further we give a gluing formula for $d(M,ρ)$.
In this note, we revisit the $Θ$-invariant as defined by R. Bott and the first author. The $Θ$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The $Θ$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $Θ$-invariant that we can define even if the cohomology group is not vanishing.
We define an invariant of rational homology 3-spheres via vector fields. The construction of our invariant is a generalization of both that of the Kontsevich-Kuperberg-Thurston invariant and that of Watanabe's Morse homotopy invariant, which implies the equivalence of these two invariants.