Laver Tables: from Set Theory to Braid Theory

Laver tables An are certain finite shelves (i.e., sets endowed with a binary operation distributive with respect to itself). They originate from Set Theory and, in spite of an elementary definition, have complicated combinatorial properties. They are conjectured to approximate the free monogenerated shelf F1, this conjecture being currently proved only under a large cardinal axiom. This talk is devoted to our dreams concerning potential braid and knot invariant constructions using Laver tables, and to some real results in this direction, such as a detailed description of 2and 3-cocycles for the An. The rich structure of the latter, as well as spectacular applications of F1 to Braid Theory, promise interesting topological consequences. 1. A Laver table is... We start with a formal presentation of the main characters of our story: Definition 1.1. ➺ A shelf is a set S endowed with a binary operation ⊲ satisfying the (left) self-distributivity condition a ⊲ (b ⊲ c) = (a ⊲ b) ⊲ (a ⊲ c). (1) ➺ The free shelf generated by a single element is denoted by F1. ➺ The Laver table An is the unique shelf ({1, 2, 3, . . . , 2 }, ⊲n) satisfying the initial condition a ⊲n 1 ≡ a+ 1 mod 2 . (2) When working modulo N , we will systematically replace the element 0 with N , which is a less conventional representative of the same class. Further, all formulas in An will only hold modulo 2 , which will be often omitted for brevity. While the first two notions regularly appear (under different names) in Low-Dimensional Topology, Set Theory and Hopf Algebra Theory, the last one is much more exotic. In this preliminary section we will discuss its origin, explain why it is well defined, and present some of its (rather astonishing) properties. This work was supported by a JSPS Postdoctral Fellowship For Foreign Researchers and by JSPS KAKENHI Grant 25·03315. 2010 Mathematics Subject Classification: 57M27, 17D99, 20N02, 55N35, 06A99.