Geometric presentations for Thompson's groups

We prove that Thompson's groups $F$ and $V$ are the geometry groups of associativity, and of associativity together with commutativity, respectively. We deduce new presentations of $F$ and $V$. These presentations lead to considering a certain subgroup of $V$ and an extension of this subgroup. We prove that the latter are the geometry groups of associativity together with the law $x(yz) = y(xz)$, and of associativity together with a twisted version of this law involving self-distributivity, respectively

Using groups for investigating rewrite systems

We describe several technical tools that prove to be efficient for investigating the rewrite systems associated with a family of algebraic laws, and might be useful for more general rewrite systems. These tools consist in introducing a monoid of partial operators, listing the monoid relations expressing the possible local confluence of the rewrite system, then introducing the group presented by these relations, and finally replacing the initial rewrite system with a internal process entirely sitting in the latter group. When the approach can be completed, one typically obtains a practical method for constructing algebras satisfying prescribed laws and for solving the associated word problem

Knots and distributive homology: from arc colorings to Yang-Baxter homology

This paper is a sequel to my essay "Distributivity versus associativity in the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011, 821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology, in particular giving a simple description of homology of biquandles.Comment: 64 pages, 29 figures; to be published as a Chapter in: "New Ideas in Low Dimensional Topology", World Scientific, Vol. 5