1,896 research outputs found
A characterization of partially dual graphs
In this paper, we extend the recently introduced concept of partially dual
ribbon graphs to graphs. We then go on to characterize partial duality of
graphs in terms of bijections between edge sets of corresponding graphs. This
result generalizes a well known result of J. Edmonds in which natural duality
of graphs is characterized in terms of edge correspondence, and gives a
combinatorial characterization of partial duality.Comment: V2: the statement of the main result has been changed. To appear in
JGT
Topological arbiters
This paper initiates the study of topological arbiters, a concept rooted in
Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological
arbiter associates a value 0 or 1 to codimension zero submanifolds of W,
subject to natural topological and duality axioms. For example, there is a
unique arbiter on , which reports the location of the essential 1-cycle.
In contrast, we show that there exists an uncountable collection of topological
arbiters in dimension 4. Families of arbiters, not induced by homology, are
also shown to exist in higher dimensions. The technical ingredients underlying
the four dimensional results are secondary obstructions to generalized
link-slicing problems. For classical links in the 3-sphere the construction
relies on the existence of nilpotent embedding obstructions in dimension 4,
reflected in particular by the Milnor group. In higher dimensions novel
arbiters are produced using nontrivial squares in stable homotopy theory.
The concept of "topological arbiter" derives from percolation and from
4-dimensional surgery. It is not the purpose of this paper to advance either of
these subjects, but rather to study the concept for its own sake. However in
appendices we give both an application to percolation, and the current
understanding of the relationship between arbiters and surgery. An appendix
also introduces a more general notion of a multi-arbiter. Properties and
applications are discussed, including a construction of non-homological
multi-arbiters.Comment: v3: A minor reorganization of the pape
Milnor Invariants for Spatial Graphs
Link homotopy has been an active area of research for knot theorists since
its introduction by Milnor in the 1950s. We introduce a new equivalence
relation on spatial graphs called component homotopy, which reduces to link
homotopy in the classical case. Unlike previous attempts at generalizing link
homotopy to spatial graphs, our new relation allows analogues of some standard
link homotopy results and invariants.
In particular we can define a type of Milnor group for a spatial graph under
component homotopy, and this group determines whether or not the spatial graph
is splittable. More surprisingly, we will also show that whether the spatial
graph is splittable up to component homotopy depends only on the link homotopy
class of the links contained within it. Numerical invariants of the relation
will also be produced.Comment: 11 pages, 5 figure
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