What is a braidoid diagram?

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.

2 thoughts on “What is a braidoid diagram?

  1. Very nice! Thank you very much for a delightful and informative talk. If anyone would like to ask a question of the speaker, please reply to this post below.

    1. Do braidoids admit an algebraic structure? In particular, is it possible to define a “braidoid group” that is analogous to the usual braid group? Thank you!

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