Stiff knots

We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are that (i) their equilibrium energy as well as the equilibrium tension for open knots depends on the type of knot as the square of the bridge number and (ii) braid localization is found to be a general feature of stiff string entanglements, while angle and knot localizations 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.