The shapes of (tangled) thickened curves.

May 24, 2023
I will present results of computer experiments on thickened–tubified–curves. A thickened curve is a curve that may be thickened to an embedded tube of unit radius, allowing us to choose a scale at which we may measure the curve’s length, a scale invariant length. A closed (tangled) thickened curve’s scale invariant length is lower bounded by the knot invariant ropelength but such curves need not be tight, they can be loosely tied. This work investigates the shapes of loosely tied thickened curves by performing deformation experiments without changing the curves’ scale invariant length. The main inspiration of this research is the role of tangling in filament–like biomaterials in solution, such as proteins existing in the aqueous environment of living cells. From this physically motivated perspective the experiment optimises the curve trajectory towards thickened curve shapes with a high degree of thermodynamic stability in solution. How can entanglement hinder or coordinate the energetically driven shape change of a thickened curve? This is joint work with Prof. Myf Evans of the University of Potsdam.