Convergence and Isotopy Type for Graphs of Finite Total Curvature

Abstract

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result for essential subarcs of a knot.

Quadrisecants give new lower bounds for the ropelength of a knot

“…We could allow the perturbation only in that case of intersection; the combing arguments of [7] show the resulting definition is equivalent. We require only that S 0 be "-close to S in the C 0 sense; it thus could be locally knotted, but in the end we care only about the homotopy class h, and not an isotopy class.…”

Isotopic equivalence by Bézier curve subdivision for application to high performance computing

“…It was proved that for homeomorphic curves, if their distance and angles between first derivatives are within given bounds, then these curves are ambient isotopic (Denne and Sullivan, 2008). We invoke this previous result.…”

“…In this section, we consider a closed Bézier curve B, and derive the ambient isotopy following Denne and Sullivan (2008, Proposition 3.1). We note that C 1,1 was previously used (Denne and Sullivan, 2008), which is true for B by Corollary 4.1. The stronger C 2 assumption invoked, below, is to ensure the singularity of the pipe surface, as previously noted in Section 2.…”

Topological Sholl Descriptors For Neuronal Clustering and Classification

“…Our next definition is adapted from ( [45], §2) who utilizes it for rectifiable curves and in the context of knot theory.…”