Computational Topology: Isotopic Convergence to a Stick Knot

TL;DR

This paper explores isotopic convergence of smooth knots to stick knots in computational topology, providing conditions for isotopic equivalence and enabling advanced computational experiments in the field.

Contribution

It extends previous convergence results to isotopic convergence, offering a priori conditions and enhancing the study of piecewise linear knots in computational topology.

Findings

Established sufficient conditions for isotopic equivalence

Demonstrated convergence through computational experiments

Facilitated further research on PL knots

Abstract

Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergence, with that discovery aided by computational experiments. Sufficient conditions to attain isotopic equivalence can be determined a priori. These sufficient conditions need not be tight bounds, providing opportunities for further optimizations. The results presented will facilitate further computational experiments on the theory of PL knots (also known as stick knots), where this theory is less mature than for smooth knots.