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

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arxiv 1409.7044 v1 pith:LC6VIMXE submitted 2014-09-24 math.GT math.AT

Jozef H. Przytycki This is my paper · ORCID
classification math.GTmath.AT
keywords homologyyang-baxtercoloringsdistributivemaththeoryalgebraicarxiv
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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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. In Search of Homology for Quasigroups of Bol-Moufang Type

    math.GR 2025-08 conditional novelty 6.0 of 10

    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.

  2. Homology of Yang-Baxter modules

    math.QA 2025-05 reject novelty 6.0 of 10

    It claims a clean algebraic decomposition of the one-term Yang-Baxter homology for sl_m R-matrices, but the main eigenspace decomposition appears incorrect for small examples.

  3. Path to homology of Yang-Baxter operators

    math.GT 2026-07 accept novelty 5.5 of 10

    A historical and personal survey of the path from knot theory and distributive homology to Yang-Baxter homology, including new homology computations for HOMFLYPT operators and a cubic face-map decomposition of the Rei...

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