On isotopy, I

n-QUASI-ISOTOPY: I. QUESTIONS OF NILPOTENCE

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link up to link homotopy can be cancelled in a (componentwise) connected sum with another link. In this paper we address the question whether some form of the ‘complexity accumulation’ property of knots holds for (piecewiselinear) links up to some stronger analogue of link homotopy, which still do...

Testing Graph Isotopy on Surfaces

We investigate the following problem: Given two embeddings G1 and G2 of the same abstract graph G on an orientable surface S, decide whether G1 and G2 are isotopic; in other words, whether there exists a continuous family of embeddings between G1 and G2. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G1 (resp., G...

Notes on the Braid Isotopy Problem

Isotopy Stability of Dynamics on Surfaces

This paper investigates dynamics that persist under isotopy in classes of orientation-preserving homeomorphisms of orientable surfaces. The persistence of periodic points with respect to periodic and strong Nielsen equivalence is studied. The existence of a dynamically minimal representative with respect to these relations is proved and necessary and sufficient conditions for the isotopy stabil...

Isotopy Convergence Theorem

When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We establish a criterion to preserve knot type under approximation that relies upon convergence in both distance and total curvature.