Stiff knots

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.

Submitted to Phys. Rev. E

Cited by in corpus (10)

• Influence of Rigidity and Knot Complexity on the Knotting of Confined Polymers
• Jamming proteins with slipknots and their free energy landscape
• Effect of bending rigidity on the knotting of a polymer under tension
• Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non- trivial contact topology
• Periodic motion of sedimenting flexible knots
• The elastic trefoil is the twice covered circle
• Complex dynamics of knotted filaments in shear flow
• Transition to coarsening for confined one-dimensional interfaces with bending rigidity
• Elastic curves and self-intersections
• Knots with equal bridge index and braid index