523 research outputs found
Colorings, determinants and Alexander polynomials for spatial graphs
A {\em balanced} spatial graph has an integer weight on each edge, so that
the directed sum of the weights at each vertex is zero. We describe the
Alexander module and polynomial for balanced spatial graphs (originally due to
Kinoshita \cite{ki}), and examine their behavior under some common operations
on the graph. We use the Alexander module to define the determinant and
-colorings of a balanced spatial graph, and provide examples. We show that
the determinant of a spatial graph determines for which the graph is
-colorable, and that a -coloring of a graph corresponds to a
representation of the fundamental group of its complement into a metacyclic
group . We finish by proving some properties of the Alexander
polynomial.Comment: 14 pages, 7 figures; version 3 reorganizes the paper, shortens some
of the proofs, and improves the results related to representations in
metacyclic groups. This is the final version, accepted by Journal of Knot
Theory and its Ramification
Levels of knotting of spatial handlebodies
If H is a spatial handlebody, i.e. a handlebody embedded in the
3-sphere, a spine of H is a graph Γ ⊂ S 3 such that H is a regular neighbour-hood of Γ. Usually, H is said to be unknotted if it admits a planar spine. This suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties. Building on this remark, we define several levels of knotting of spatial handlebodies, and we provide a complete description of the relationships between these levels, focusing our attention on the case of genus 2. We also relate the knotting level of a spatial handlebody H to classical topological properties of its complement M = S 3 \ H, such as its cut number. More precisely, we show that if H is not highly knotted, then M
admits special cut systems for M , and we discuss the extent to which the converse implication holds. Along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies. These obstructions are based on recent quandle–coloring invariants for spatial handlebodies, on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links, on the analysis of appropriate coverings of handlebody complements, and on the study of the classical Alexander elementary ideals of their fundamental groups
The Bipartite Swapping Trick on Graph Homomorphisms
We provide an upper bound to the number of graph homomorphisms from to
, where is a fixed graph with certain properties, and varies over
all -vertex, -regular graphs. This result generalizes a recently resolved
conjecture of Alon and Kahn on the number of independent sets. We build on the
work of Galvin and Tetali, who studied the number of graph homomorphisms from
to when is bipartite. We also apply our techniques to graph
colorings and stable set polytopes.Comment: 22 pages. To appear in SIAM J. Discrete Mat
Three geometric applications of quandle homology
In this paper we describe three geometric applications of quandle homology.
We show that it gives obstructions to tangle embeddings, provides the lower
bound for the 4-move distance between links, and can be used in determining
periodicity of links.Comment: 16 pages, 13 figure
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