903 research outputs found
Counting the spanning trees of the 3-cube using edge slides
We give a direct combinatorial proof of the known fact that the 3-cube has
384 spanning trees, using an "edge slide" operation on spanning trees. This
gives an answer in the case n=3 to a question implicitly raised by Stanley. Our
argument also gives a bijective proof of the n=3 case of a weighted count of
the spanning trees of the n-cube due to Martin and Reiner.Comment: 17 pages, 9 figures. v2: Final version as published in the
Australasian Journal of Combinatorics. Section 5 shortened and restructured;
references added; one figure added; some typos corrected; additional minor
changes in response to the referees' comment
The Edge Slide Graph of the 3-cube
The goal of this paper is to study the spanning trees of the 3-cube by understanding their \emph{edge slide graph}. A spanning tree of a graph is a minimal set of edges that connects all vertices. An edge slide occurs in a spanning tree of the 3-cube when a single edge can be slid across a 2-dimensional face to form another spanning tree. The edge slide graph is the graph whose vertices are the spanning trees, with an edge between two vertices if the spanning trees are related by a single edge slide. This report completely determines the edge slide graph of the 3-cube. The edge slide graph of the 3-cube has twelve components isomorphic to the 4-cube, and three other components, mutually isomorphic, with 64 vertices each. The main result is to determine the structure of the three components that each have 64 vertices and we also describe their symmetries. Some partial results on the 4-cube are also provided
Combinatorial cohomology of the space of long knots
The motivation of this work is to define cohomology classes in the space of
knots that are both easy to find and to evaluate, by reducing the problem to
simple linear algebra. We achieve this goal by defining a combinatorial graded
cochain complex, such that the elements of an explicit submodule in the
cohomology define algebraic intersections with some "geometrically simple"
strata in the space of knots. Such strata are endowed with explicit
co-orientations, that are canonical in some sense. The combinatorial tools
involved are natural generalisations (degeneracies) of usual methods using
arrow diagrams.Comment: 20p. 9 fig
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