The paper defines ∂2 and ∂3 boundary maps for Bol-Moufang quasigroups and computes H1 and H2 for the distinguishing examples of Phillips and Vojtechovsky.
Knots and distributive homology: from arc colorings to Yang-Baxter homology
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abstract
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.
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In Search of Homology for Quasigroups of Bol-Moufang Type
The paper defines ∂2 and ∂3 boundary maps for Bol-Moufang quasigroups and computes H1 and H2 for the distinguishing examples of Phillips and Vojtechovsky.