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This paper is a sequel to my essay "Distributivity versus associativity inthe homology theory of algebraic structures" Demonstratio Math., 44(4), 2011,821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arccolorings and survey associative and distributive magmas and their homologywith relation to knot theory. We outline potential relations to Khovanovhomology and categorification, via Yang-Baxter operators. We use here the factthat Yang-Baxter equation can be thought of as a generalization ofself-distributivity. We show how to define and visualize Yang-Baxter homology,in particular giving a simple description of homology of biquandles.

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