4,474 research outputs found
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
Generalized algebra within a nonextensive statistics
By considering generalized logarithm and exponential functions used in
nonextensive statistics, the four usual algebraic operators : addition,
subtraction, product and division, are generalized. The properties of the
generalized operators are investigated. Some standard properties are preserved,
e.g., associativity, commutativity and existence of neutral elements. On the
contrary, the distributivity law and the opposite element is no more universal
within the generalized algebra.Comment: 11 pages, no figure, TeX. Reports on Mathematical Physics (2003), in
pres
Trimness of Closed Intervals in Cambrian Semilattices
In this article, we give a short algebraic proof that all closed intervals in
a -Cambrian semilattice are trim for any Coxeter
group and any Coxeter element . This means that if such an
interval has length , then there exists a maximal chain of length
consisting of left-modular elements, and there are precisely join- and
meet-irreducible elements in this interval. Consequently every graded interval
in is distributive. This problem was open for any
Coxeter group that is not a Weyl group.Comment: Final version. The contents of this paper were formerly part of my
now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure
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