Stiff Knots : Relating topology to geometry and mechanics(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)

Abstract

We analyze the energy and the geometry of closed knotted strings. We focus on the properties of the ground state (the state of lowest energy). We point out the important role of the the bridge number n. Indeed, the energy of the ground state exhibits upper and lower bounds which both scale as n^2. The geometry of the knot in the ground state is found explicitly in some special cases. Finally, we mention some additional results on the role of torsional elasticity, on the multiplicity of contact points and on the size of the knot.