AIMR Math Group Online Seminar
#15 FY2023
Date : February. 13th (Tue) 2024 16:00-17:00
Venue:Zoom
Speaker: Toky Andriamanalina (University of Potsdam)
Title: Knot diagrams for 3-periodic entanglements
abstract:
Polymers, DNA origami crystals, and many other biological and chemical structures present features of entanglement, arranging in a 3-periodic fashion. As the topology of those structures have influence on their physical properties, some knot invariants have been extended into measures, such as the periodic linking number or the periodic Jones polynomials, to quantify the entanglement. This project aims to give a new mathematical diagrammatic description based on Knot theory for 3-periodic entangled structures, that we call 3-periodic tangles or TP tangles. To do so, we project a unit cell of the TP tangle, which is a 3-torus, onto a 2-torus represented as a square with identified edges. The new diagrams require a set of new moves added to the three usual Reidemeister moves, as well as other notions of equivalence, that we show to preserve ambient isotopies of 3-periodic tangles.
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