Speakers:
Title: From the Kauffman bracket (linear) skein modules via quadratic skein modules to cubic skein modules and beyond
Abstract: The study of linear and quadratic skein modules over the last thirty-seven years has lead to a very rich skein theory that is connected to many disciplines of mathematics and physics, such as algebraic geometry, hyperbolic geometry, Topological Quantum Field Theories (TQFT), and statistical mechanics. I start my presentation by outlining the progress in study of linear and quadratic skein modules with a stress on the Kauffman bracket skein modules and Witten conjecture. There is, however, another class of skein modules with more parameters than the linear and quadratic cases which, save for a few exceptions, have been largely neglected until now. The cubic skein module is the first object in this class which awaits exploration. I devote the last part of my talk to analysis of cubic (and beyond) skein modules. This is a joint work with Mathathoners VIII (Rhea Palak Bakshi, Anthony Christiana, Ali Guo, Dionne Ibarra, Gabriel Montoya Vega, Sujoy Mukherjee and Xiao Wang).
Abstract: Tribrackets are sets with a ternary operation satisfying axioms coming from the Reidemeister moves in knot theory. In particular, finite tribrackets define invariants of knots and links in terms of tribracket homsets, which can be computed from diagrams. By the way, S. Nelson introduced an algebraic structure known as biquasile, which is used to define oriented link invariants via colorings of certain graphs obtained from oriented link diagrams. Biquasile colorings can be understood in terms of ternary quasigroup colorings, and these Boltzmann weights can be understood as enhancement by cocycles in ternary quasigroup cohomology. In this talk, we define invariants of links, surface-links, pseudoknots, and singular knots. This is a joint work with S. Nelson and S. Jeong.
Title: Homology theory of racks of finite ranks and link invariants.
Abstract: Quandle homology was defined by CJKLS, from rack homology defined by FRS, as a quotient by degenerate subcomplex coming from idempotency. There has been many variations of this homology theory. In this talk, we will discuss an early (2010) joint work with Sam regarding homology theory of racks with finite ranks. Let N be the rank of a rack. Then colorings of link diagrams by the rack are preserved by the N-phone chord moves and thus give a counting invariant of links. Nontrivial 2-cocycles are used to construct an enhancement of this counting invariant.
Abstract: In knot theory, algebraic structures are commonly utilized to distinguish knots. This talk will delve into a particular algebraic structure designed for singular knots, known as a singquandle. We will use the singquandle algebraic structure to explore the singquandle counting invariant for singular knots. Subsequently, we will introduce different methods to enhance the singquandle counting invariant, aiming to derive more robust invariants for singular knots.
Title: Categorification, quivers, and biquandle brackets
Abstract: This talk will explore the confluence of Sam Nelson's and my research trajectories, focusing on topics such as categorification, quivers, and the Khovanov-like categorification of the biquandle bracket.
Title: Triple point numbers of some 2-twist spun knots
Abstract: This is joint work with Seonmi Choi, Xiao Wang, and Seung Yeop Yang, Fox's examples 12 and example 15 are known to be twist-spins of certain rational tangles by results of Litherland and Kanenobu. Example 15 is in fact a family of examples. The movies that were suggested by Fox's construction seem to have the fewest number of Reidemeister type III moves among all movies of the represented $2$-knot. In example 12, Shin Satoh and Akiko Shima have shown that this number of type III moves is minimal. Satoh has also demonstrated the same thing for the first member of the family in Example 15. We examine the next element of example 12.