Isotopic Convergence Theorem

TL;DR

This paper establishes a criterion ensuring the preservation of knot type when approximating space curves, based on pointwise convergence and total curvature convergence, which is significant for computer graphics and visualization.

Contribution

The paper introduces a new criterion for preserving knot type during space curve approximation, combining pointwise convergence with total curvature convergence.

Findings

Provides a criterion for knot type preservation in curve approximation

Links convergence in total curvature with topological invariance

Applicable to computer graphics and visualization tasks

Abstract

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 pointwise convergence and convergence in total curvature.