Emanuele Zappala (University of South Florida)
In this talk I recall the definitions of shelf/rack/quandle and their cohomology theory. I also give the construction of cocycle invariant of links, due to Carter, Jelsovsky, Kamada, Langford and Saito (Trans. Amer. Math. Soc. 355 (2003), 3947-3989). Then, I introduce higher arity self-distributive structures and show that an appropriate diagrammatic interpretation of them is suitable to define a ternary version of the cocycle invariant for framed link invariants, via blackboard framings. I discuss the computation of cohomology of ternary structures, as composition of mutually distributive operations, and a cohomology theory of certain ternary quandles called group heaps. Furthermore I mention a categorical version of self-distributivity, along with examples from Lie algebras and Hopf algebras, and the construction of ribbon categories from ternary operations that provide a quantum interpretation of the (ternary) cocycle invariant.
References:
Elhamdadi, Mohamed, Masahico Saito, and Emanuele Zappala. “Higher Arity Self-Distributive Operations in Cascades and their Cohomology.” arXiv preprint arXiv:1905.00440 (2019).
Elhamdadi, Mohamed, Masahico Saito, and Emanuele Zappala. “Heap and Ternary Self-Distributive Cohomology.” arXiv preprint arXiv:1910.02877 (2019).
Zappala, Emanuele. “Non-Associative Algebraic Structures in Knot Theory.” (2020).
Thank you, Emanuele, for a fascinating talk! To ask a question of our speaker, please reply to this question below.
Very interesting! Now that you have an invariant of framed links, have you thought about how it behaves under Kirby moves or Rolfsen moves? For example, it would be cool if you could get 3-manifold invariants from your construction. Thank you!