Alexander Theorem

Some algebraic structures for flat virtual links

| chrisman.76

David Freund (Harvard University)

Flat virtual links are generalizations of curves on surfaces that have come out of Kauffman’s virtual knot theory. In particular, we consider the curves up to homotopy and allow the supporting surface to change via the addition or removal of empty handles. Under this equivalence, a flat virtual link obtains a completely combinatorial structure. Analogous to the classical problem of finding the minimal number of intersection points between two curves, we can ask for the minimal number of intersection points between components of a flat virtual link. By moving between geometric and combinatorial models, we develop generalizations of the Andersen-Mattes-Reshetikhin Poisson bracket and compute it for infinite families of two-component flat virtual links using a generalization of Henrich’s singular based matrix for flat virtual knots. Throughout the talk, we emphasize the motivation behind different constructions.

What is a braidoid diagram?

| chrisman.76

Neslihan Gügümcü (Izmir Institute of Technology and University of Goettingen)

In this talk we first review the basics of the theory of knotoids introduced by Vladimir Turaev in 2012 [1]. A knotoid diagram is basically an open-ended knot diagram with two open endpoints that can lie in any local region complementary to the plane of the diagram. The theory of knotoids extends the classical knot theory and brings up some interesting problems and features such as the height problem [1,3] and parity notion and related invariants such as off writhe and parity bracket polynomial [4]. It was a curious problem to determine a “braid like object” corresponding to knotoid diagrams. The second part of this talk is devoted to the theory of braidoids, introduced by the author and Sofia Lambropoulou [2]. We present the notion of a braidoid and analogous theorems to the classical Alexander Theorem and the Markov Theorem, that relate knotoids/multi-knotoids in the plane to braidoids.

[1] V.Turaev, Knotoids, Osaka J.of Mathematics 49 (2012), 195–223. https://arxiv.org/abs/1002.4133

[2] N.Gugumcu and S.Lambropoulou, Braidoids, to appear in Israel J.of Mathematics, https://arxiv.org/abs/1908.06053

[3] N.Gugumcu and L. Kauffman, New Invariants of Knotoids, European J.of Combinatorics, (2017), 65C, 186-229, https://arxiv.org/abs/1602.03579

[4] The Guassian parity and minimal diagrams of knot-type knotoids, submitted.