Self-contact preventing energy for tubular rods

Abstract

We introduce a generalization of Mobius energy for knots to an energy functional for tubular neighbourhoods of closed inextensible curves. We prove the continuity of the energy and its boundedness for physically admissible tubes without self-contact and in particular for the torus. The functional allows to distinguish isotopy classes of the centerline through its physical inspiration to a self-repulsive electrostatic energy. If the tube has zero thickness, O'Hara's functional is recovered. Finally, a discussion on the possible exponents in the functional is carried out.

Figures (4)

• Figure 1: Representation of the map $p$ from the set $\Omega$ to $p(\Omega)$. A point $\Tilde{P}$ is mapped into $P\in p(\Omega)$, defined as above. In particular, here $P\in \operatorname{bd}T_r$.
• Figure 2: Representation of the desired subdivision: every tubular neighbourhood is identified with its centerline $\gamma$. The space of the possible knot types of $\gamma$ is partitioned into isotopy classes, separated by walls (blue lines) that represent configurations with singular knots as centerlines. Each class is divided into two regions: the green one, with physically admissible neighbourhoods and the orange one, with interpenetration of matter. The black squares represent the configurations with self-contact on the boundary and without interpenetration, so physically consistent. We show an example for each mentioned case in the class of the unknot, and an example of the trefoil knot. The paraboloid represents the ideal behaviour of the energy functional that we are looking for, analogous to the one proposed by O'Hara for knots, with a minimum in each class and the circular centerline as global minimizer.
• Figure 3: As an example, we plot an arc of helix described as the curve $\gamma$, with $\gamma(\xi)=(R\cos(\xi/\lambda),R\sin(\xi/\lambda),a\xi/\lambda)$, $\xi \in [0,7/2]$, $R=2,a=0.5,\lambda=\sqrt{R^2+a^2}$. We show the tubular neighbourhood $T_r$ around $\gamma$ with radius $r=0.5$, and we highlight the cross-sections corresponding to $s$ and $t$ and the Serret-Frenet frame in $\gamma(s)$ and $\gamma(t)$, with $s=0$ and $t=2$. Chosen the points $X(0,5\pi/6)$ and $Y(2,\pi/2)$, we have $Z_1(2,5\pi/6), Z_2(0,\pi/2)$ in the figure. We have the constant curvature $\kappa=R/\lambda^2\sim 0.47$ and the torsion $\tau=a/\lambda^2\sim 0.118$. In red, the two arcs of meridians and in yellow and brown the two parallels $P(\theta)$ and $P(\varphi)$. Finally, we show the decomposition of $P(\varphi)$ in a light blue arc $P^{\boldsymbol{n},\boldsymbol{b}}$ on the boundary of $\mathcal{A}(t)$ that has length $\hat{l}_P^{\boldsymbol{n},\boldsymbol{b}}=r \tau (t-s)$ and a purple curve $P(\varphi)^\mathbf{t}$ orthogonal to the cross-section $\mathcal{A}(t)$ with length $|t-s-r (t-s) \cos \varphi \kappa|$ that in our case equals $t-s$ because $\varphi=\pi/2$.
• Figure 4: (a) The torus with the four points $X,Y,Z_1,Z_2$ with the parameters $R=2,r=1,u=\frac{11\pi}{6},v=\frac{\pi}{3},\theta=\frac{\pi}{3},\varphi=\frac{5\pi}{6}$. In red the two arcs of meridians considered in ${d^*}^2$, in yellow the parallel at angle $\varphi$ and in brown the parallel at angle $\theta$. The four points belong to the same plane $\alpha: 1.43x+0.38y+3.92z-4.96=0$ plotted in light blue. (b) The construction of the isosceles trapezoid in the plane $\alpha$; we use Pythagoras' theorem to find an equality involving the two basis $a$ and $b$, the edges $c$ and the diagonal $d$.

Theorems & Definitions (30)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Proposition 2.7
• Theorem 2.8
• Corollary 2.9
• Definition 3.1