# The moduli space of two-convex embedded tori

## TL;DR

This paper studies the topology of the space of two-convex embedded tori in Euclidean space, showing path-connectedness in higher dimensions and a correspondence with knot classes in dimension two, using a specialized mean curvature flow.

## Contribution

It establishes the topological structure of the moduli space of two-convex embedded tori, including path-connectedness and knot class correspondence, via a novel mean curvature flow technique.

## Key findings

- The moduli space is path-connected for dimensions n ≥ 3.
- In dimension 2, the connected components correspond to knot classes.
- A new mean curvature flow with surgery approach preserves topology and convexity.

## Abstract

In this short article we investigate the topology of the moduli space of two-convex embedded tori $S^{n-1}\times S^1\subset \mathbb{R}^{n+1}$. We prove that for $n \geq 3$ 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.