Topology Triple-Header!

This triple-header of topology talks will include three speakers:

First, Hyeran Cho from The Ohio State University will speak about Derivation of Schubert normal forms of 2-bridge knots from $(1,1)$-diagrams.

In this talk, we show that the dual (1, 1)-diagram of a (1, 1)-diagram (a.k.a. a two pointed genus one Heegaard diagram)
D(a, 0, 1, r) with 1 ≤ r < 2a + 1 and gcd(2a + 1, r) = 1 is given by D(1/2r, 0, 2a+1-1/r, 1/r) when 1/r is even and by D((2a+1−r)/2, 0, r −1, r −1) otherwise,  where 1/r is the multiplicative inverse of r modulo 2a + 1. We also present explicitly how to derive a Schubert normal form of a 2-bridge knot from the dual (1, 1)-diagram of D(a, 0, 1, r) using weakly K−reducibility of (1, 1)-
decompositions.

Second, Suhyeon Jeong from Pusan National University will speak about Psybrackets, Singular Knots and Pseudoknots.:

In 2010, a pseudodiagram was introduced by Ryo Hanaki. A pseudodiagram is a knot or link diagram where we ignore over/under information at some crossings of the diagram. This definition is motivated by applications in molecular biology such as modeling knotted DNA, where data often comes inconclusive with respect to which crossing it represents. In 2012, Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten, and Ljiljana Radvić extended this idea to a pseudoknot and pseudolink. A pseudoknot (or pseudolink ) is an equivalence class of pseudodiagrams modulo pseudo-Reidemeister moves. In this talk, we would like to introduce a psybracket consisting of two maps <, , > c , <, , > p : X × X × X → X satisfying some axioms derived from pseudo-Reidemeister moves. By using this, we define an invariant, called the psybracket counting invariant, of oriented singular knots and links and pseudolinks. This is a joint work with Jieon Kim and Sam Nelson.

Finally, Minju Seo from Pusan National University will speak about Quandle coloring quivers of surface-links.: 

In 2018, K. Cho and S. Nelson introduced the quandle coloring quiver of an oriented knot or link diagram, which is a quiver structure on the set of quandle colorings of a knot or link diagram. Also, they gave a new invariant, called the in-degree quandle quiver polynomial, from the quiver structure. A surface-link is a closed 2-manifold smoothly embedded in R 4 or S 4 . A surface-link can be presented by a marked graph diagram with specific condition, and a marked graph diagram is a generalization of a knot or link diagram. In this talk, we introduce a quiver structure on the set of quandle colorings of a marked graph diagram, and compute the in-degree quandle quiver polynomials of some marked graph diagrams. This is a joint work with J. Kim and S. Nelson.