Quantum Topology / Hopf Algebra Seminar
Jonathan Schneider
UIC
What is a Rotweldid Knot?
Abstract: This series of talks represents joint work with Eiji Ogasa and Lou Kauffman. Shin Satoh provided a topological model for the diagrammatic theory of Welded knots. Specifically, he described a map from knot diagrams to toral surfaces which respects Welded-equivalence, so equivalent diagrams are modeled by isotopic toral surfaces. However, Satoh was unable to determine whether this model was classifying for Welded knot theory— whether inequivalent Welded knot diagrams might nonetheless be modeled by isotopic toral surfaces. The question of whether Satoh’s model is classifying remains open as of 2017. Colin Rourke claimed to provide another model for Welded knot theory, based on Satoh’s model but “strengthened”. In his proposal, Rourke adorned Satoh’s toral surfaces with a fiber-structure. Under this construction, inequiv- alent knot diagrams never map to fiberwise-isotopic toral surfaces, so Rourke’s model— if it is a valid model at all— is a classifying model for the theory. Unfortunately, Rourke’s construction is not invariant of Welded knot type , thus it is not a valid model for Welded knot theory. This is Main Theorem 1 of the present paper. We prove the theorem by exhibiting two knot diagrams which are Welded-equivalent, but whose Rourke-models are not fiberwise-isotopic. 1. Rourke’s construction is thus useless as a model for Welded knot theory. However, the theory can be modified into a new diagrammatic theory, which we call Rotweldid knot theory, by omitting the single Welded Reidemeister move where Rourke’s invariance proof failed. The invariance of Rourke’s construction as a model for Rotweldid knot theory is our Main Theorem 2. The proof is the same as Rourke’s erroneous proof for Weldid, with the bad part excised. Not only is Rourke’s construction a valid model for Rotweldid knot theory, it also happens to be a classifying model. This is our Main Theorem 3.
Thursday October 5, 2017 at 3:00 PM in SEO 636