The Moduli Space of Two-Convex Embedded Tori

In this short article we investigate the topology of the moduli space of two-convex embedded tori S^(n-1)×S^1⊂R^(n+1). We prove that for n≥3 this moduli space is path-connected, and that for n=2 the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article (arXiv:1607.05604) where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topology, embeddedness, as well as two-convexity.