Total curvature of curves in the C¹-closure of knot classes with finite, isolated self intersections

• Totalkrümmung von Kurven im C¹-Abschluss von Knotenklassen mit endlichen, isolierten Selbstschnitten

Wacker, Elisabeth; von der Mosel, Heiko (Thesis advisor); Wagner, Alfred (Thesis advisor)

Aachen : RWTH Aachen University (2021, 2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

The Fáry-Milnor Theorem states that the total curvature of a knot gamma, which is a simple closed curve, is bounded from below by 2 pi times the bridge number br of the knot class of gamma. For knot classes K with br(K)=2, this bound also holds for curves in the C^1-closure of K, as proven by Gerlach et al. using the existence of alternating quadrisecants, a result of Denne. A general version of the Fáry-Milnor Theorem could turn out to be useful to characterise elastic knots for more general knot classes than treated by Gerlach et al.. In this thesis, we prove an extended Fáry-Milnor Theorem: For a curve gamma in the C^1-boundary of a knot class K with only finitely many, isolated self intersections we bound the total curvature from below by 2 pi br(K). By construction, there is a sequence (gamma_k)_{k in N} subset K approximating gamma in C^1. We first prove the statement for all gamma with purely transversal self intersections. We achieve this with ambiently isotopic transformations of the gamma_k such that the total curvature of the transformed sequence approximates the total curvature of the limit curve gamma. In particular, we apply the technique of a foliation of disjoint, compact, convex and planar cross sections from von der Mosel. Moreover, we introduce new isotopies being explicitly defined by double-cone constructions. Next, we extend the result to limit curves with non-transversal self intersections. To that end, we introduce the concept of ``weaving lines'' to which we can push gamma_k very closely by ambient isotopies. We provide an explicit construction for these weaving lines, which depend on the complexity of the self intersections of the limit curve gamma. By this procedure, we are able to control the total curvature and prove the extended Fáry-Milnor Theorem.