Entanglements & Applications

9:30-10:30: Toky Andriamanalina

Title: Untangling 3-periodic entanglements of filaments and nets

Abstract: Entanglements of curves and nets can used to describe various biological and chemical structures, such as coordination polymers, liquid crystals, or DNA origami crystals. We recently developed new diagrammatic descriptions of 3-periodic entanglements. These new diagrams are drawn out of a projection along one axis of a unit cell of a 3-periodic structure. By using these diagrams, we define the notion of untangling number for 3-periodic structures, which is a measure of complexity of the entanglement.
Thanks to this, it is now possible to characterise the least tangled structures that we call ground states, and in particular we show that the rod packings are the generic ground states of entanglements of curves.

10:30-11:00: coffee break

11:00 - 12:00: Stephen Hyde

Title: Tangles... and untangles

Abstract: Knots, braids, links, self-entangled nets, multiple catenated infinite nets... are examples of what we call, simply, “tangles”. They are relevant to molecular-scale (bio)materials, from duplexed ssRNA to metal-organic frameworks.

We are interested in understanding:
1.Which tangles are “simple”?
2.How tangled is a tangle!?

Our tangle toolkit is a simple one: we assemble helices into networks, allowing a broad spectrum of tangles to be built, from knots to tangled nets. Interesting “simple” tangles are entanglements of the edges of Platonic polyhedra [1] and entangled 2-periodic nets [2].
A proposed answer to point 2. above will be discussed. if there is time.
The ideas are at present largely unpublished, and being working into a book to be published, we hope, in late 2025 [3].

13:00 - 14:00: Myfanwy Evans

Title: Can solvents tie knots? Helical folds of biopolymers in liquid environments.

Abstract: Using a simulation technique based on the morphometric approach to solvation, we performed computer experiments which fold a short open flexible tube, modelling a biopolymer in aqueous environments, according to the interaction of the tube with the solvent alone. We find an array of helical geometries that self-assemble depending on the solvent conditions, including symmetric double helices where the strand folds back on itself and overhand knot motifs. Interestingly these shapes—in all their variety—are energetically favoured over the optimal helix. By differentiating the role of solvation in self–assembly our study helps illuminate the energetic background scenery in which all soluble biomolecules live.

This event is organized with the Interdisciplinary Math Study Group.