From Flow Spines to Virtual Knot Diagrams: A Risandle Approach to 3-Manifold Invariants

Every oriented closed 3-manifold admits a flow spine, that is, a spine in a “good” position with respect to a given nonsingular flow. A flow spine can be represented by a virtual knot diagram, where equivalence is defined through a family of local moves different from the classical Reidemeister moves. In this talk, we introduce a modified version of the quandle algebra, originally useful in knot theory, to define a new notion of “coloring” for closed 3-manifolds. This coloring yields a topological invariant of 3-manifolds. Furthermore, I will explain how this invariant is related to the fundamental group of the manifold. The correspondence between colorings and group representations will be illustrated concretely using the Poincaré homology sphere as an example. This work is a joint project with Ippei Ishii and Takuji Nakamura.