On the Recognition of Tori Embedded in R3

Abstract

Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.