Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non- trivial contact topology

TL;DR

This paper develops a mathematical model using matched asymptotic expansions to analyze the self-contact problem of elastic knots, specifically trefoil and cinquefoil, under tension and twist, revealing contact topology and effects of twist.

Contribution

It introduces a novel asymptotic approach to solve the nonlinear self-contact problem for elastic knots without prior assumptions on contact regions.

Findings

Contact set topology includes an interval and isolated points.

Applied twist influences the equilibrium configuration.

Method applicable to various knot topologies.

Abstract

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The problem is formulated as a non-linear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium.