631 research outputs found
Online and quasi-online colorings of wedges and intervals
We consider proper online colorings of hypergraphs defined by geometric
regions. We prove that there is an online coloring algorithm that colors
intervals of the real line using colors such that for every
point , contained in at least intervals, not all the intervals
containing have the same color. We also prove the corresponding result
about online coloring a family of wedges (quadrants) in the plane that are the
translates of a given fixed wedge. These results contrast the results of the
first and third author showing that in the quasi-online setting 12 colors are
enough to color wedges (independent of and ). We also consider
quasi-online coloring of intervals. In all cases we present efficient coloring
algorithms
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
- …