Quantum Topology Seminar
Jonathan Schneider
Realistic Crossing Data for Curves in the Plane
Abstract: (Continued from last week.)
When does a curve in R² with crossing data lift to a knot in R³, or, more generally, to a fiberwise toral surface in R²×R²? I propose necessary and sufficient conditions. I consider three cases: 1. Generic curves, which form the basis of familiar knot diagrams. No restrictions are necessary on crossing data for the static curve; however, a homotopy of the curve must carry the crossing data continuously and avoid "cyclic crossings". 2. Cellular curves, where the curve is a finite cellular map. Here we additionally require that the static curve itself carries crossing data continuously from point to point and avoids cyclic crossings. 3. General curves. Here, the "continuity" restriction of the first two cases is inadequate. A stronger pair of conditions, which I call "monotonicity and stability", is necessary.
Thursday February 18, 2021 at 12:00 PM in Zoom