Elastic Knots

• Applicant: Professor Dr. Philipp Reiter
• Subject Area: Mathematics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 289032105

The overall theme of this project is to characterize equilibrium shapes of knotted loops made of springy wire with respect to different models.In a first simple setting, we assume that the behavior of these knotted loops is only driven by the bending energy of the centerline, neglecting other forces such as extension, torsion, or shear. This means that the loop will attain an equilibrium corresponding to a minimum of the bending energy within its isotopy class. Regularization by the reciprocal of thickness acts like a choice criterion which singles out one solution from the potentially infinite family of bending energy minimizers. The latter solution will be referred to as an elastic knot. We aim at characterizing general elastic torus knots which requires a generalization of the Fáry-Milnor theorem.Later on we will include torsional effects in our model, based on the special Cosserat theory of inextensible and unshearable rods. This leads to twisted elastic knots. We will show that twisted elastic trefoils consist of two round circles tangentially meeting in one point and extend this result to other knot classes. In contrast to most approaches in the literature, we will be able to derive analytically rigorous results without initial assumptions on the geometry of the problem.The third part of this project comprises the development of a numerical scheme for visualization purposes. State of the art is simulated annealing which is quite robust but inefficient. We aim at exploiting the structure of the problem in order to derive an efficient algorithm based on the gradient-flow that allows for performing simulations of both elastic and twisted elastic knots.

• DFG Programme: Research Grants