Knots and Distributive Homology: From Arc Colorings to Yang–Baxter Homology
This paper has its roots in two series of talks I gave: in Russia (Lomonosov Moscow State University, May 29–June 1, 2012), where the visualization of Fig. 8 was observed, Korea (TAPU Workshop on Knot Theory, July 23–27, 2012), and in a talk at Oberwolfach Conference (June 3–9, 2012). The short version of this paper was published in Oberwolfach Proceedings. While I keep novelty of the talks (many new ideas were crystallized then), I added a lot of supporting material so the paper is mostly self-sufficient. I kept also, to some extent, the structure of talks; it may lead to some repetitions but I hope it is useful for a reader.
The following sections are included:
Introduction
Monoid of Binary Operations
Homology of Magmas
Group Homology of a Semigroup
Hochschild Homology of a Semigroup and an Algebra
Homology of Distributive Structures
Bloh–Leibniz–Loday Algebra
Semigroup Extensions and Shelf Extensions
Degeneracy for a Weak and Very Weak Simplicial Module
Degeneracy for a Weak Simplicial Module
Degeneracy for a Very Weak Simplicial Module
From Distributive Homology to Yang–Baxter Homology
Geometric Realization of Simplicial and Cubic Sets
Higher Dimensional Knot Theory Mn → ℝn+2
Acknowledgments
References
