Month: October 2020

Classifying small virtual skein theories

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Joshua Edge (Denison University)

A skein theory for the virtual Jones polynomial can be obtained from its original version with the addition of a virtual crossing that satisfies the virtual Reidemeister moves as well as a naturality condition. In general, though, knot polynomials will not have virtual counterparts. In this talk, we classify all skein-theoretic virtual knot polynomials with certain smallness conditions. In particular, we classify all virtual knot polynomials giving non-trivial invariants strictly smaller than the one given by the Higman-Sims spin model by classifying the planar algebras associated with them. This classification includes a family of skein theories coming from \text{Rep}(O(2)) with an interesting braiding. This talk is given in memory of Vaughan Jones.

Topological Deep Learning

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Mustafa Hajij (Santa Clara University)

In this work, we introduce topological deep learning, a formalism that is aimed at two goals (1) introducing topological language to deep learning for the purpose of utilizing the minimal mathematical structures to formalize problems that arise in a generic deep learning problem and (2) augment, enhance and create novel deep learning models utilizing tools available in topology. To this end, we define and study the classification problem in machine learning in a topological setting. Using this topological framework, we show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task.  To demonstrate these results, we provide example datasets and show how they are acted upon by neural nets from this topological perspective.

References:

Mustafa Hajij, Kyle Istvan, “A topological framework for deep learning”, https://arxiv.org/abs/2010.00743

Mustafa Hajij, Kyle Istvan, Ghada Zamzmi, “Cell complex neural networks”, https://arxiv.org/abs/2010.00743

Unknotting with a single twist

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Samantha Allen (Dartmouth)

Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. Using tools such as classical invariants and invariants arising from Heegaard Floer theory, we give obstructions for a knot to be unknotted with a single twist of a given linking number. In this talk, I will discuss some of these obstructions, their implications (especially for alternating knots), many examples, and some unanswered questions. This talk is based on joint work with Charles Livingston.