Stiff knots

TL;DR

This paper investigates the geometry and mechanics of stiff knotted strings, revealing how their equilibrium energy depends on knot type and identifying characteristic shapes and behaviors through simulations and experiments.

Contribution

It introduces the dependence of equilibrium energy on the square of the bridge number and uncovers braid localization as a key feature of stiff string entanglements.

Findings

Equilibrium energy scales with the square of the bridge number.

Braid localization is a universal feature in stiff string knots.

Identified three main equilibrium shapes, including a circular braid.

Abstract

We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are: (i) Their equilibrium energy as well as the equilibrium tension for open knots depend on the type of knot as the square of the bridge number; (ii) Braid localization is found to be a general feature of stiff strings entanglements, while angles and knot localization are forbidden. Moreover, we identify a family of knots for which the equilibrium shape is a circular braid. Two other equilibrium shapes are found from Monte Carlo simulations. These three shapes are confirmed by rudimentary experiments. Our approach is also extended to the problem of the minimization of the length of a knotted string with a maximum allowed curvature.