883 research outputs found
Separability and the genus of a partial dual
Partial duality generalizes the fundamental concept of the geometric dual of
an embedded graph. A partial dual is obtained by forming the geometric dual
with respect to only a subset of edges. While geometric duality preserves the
genus of an embedded graph, partial duality does not. Here we are interested in
the problem of determining which edge sets of an embedded graph give rise to a
partial dual of a given genus. This problem turns out to be intimately
connected to the separability of the embedded graph. We determine how
separability is related to the genus of a partial dual. We use this to
characterize partial duals of graphs embedded in the plane, and in the real
projective plane, in terms of a particular type of separation of an embedded
graph. These characterizations are then used to determine a local move relating
all partially dual graphs in the plane and in the real projective plane
Partial duals of plane graphs, separability and the graphs of knots
There is a well-known way to describe a link diagram as a (signed) plane
graph, called its Tait graph. This concept was recently extended, providing a
way to associate a set of embedded graphs (or ribbon graphs) to a link diagram.
While every plane graph arises as a Tait graph of a unique link diagram, not
every embedded graph represents a link diagram. Furthermore, although a Tait
graph describes a unique link diagram, the same embedded graph can represent
many different link diagrams. One is then led to ask which embedded graphs
represent link diagrams, and how link diagrams presented by the same embedded
graphs are related to one another. Here we answer these questions by
characterizing the class of embedded graphs that represent link diagrams, and
then using this characterization to find a move that relates all of the link
diagrams that are presented by the same set of embedded graphs.Comment: v2: major change
3nj Morphogenesis and Semiclassical Disentangling
Recoupling coefficients (3nj symbols) are unitary transformations between
binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum
operators. They have been used in a variety of applications in spectroscopy,
quantum chemistry and nuclear physics and quite recently also in quantum
gravity and quantum computing. These coefficients, naturally associated to
cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and
analytical features that make them fashinating objects to be studied on their
own. In this paper we develop a bottom--up, systematic procedure for the
generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and
algebraic methods. We provide also a novel approach to the problem of
classifying various regimes of semiclassical expansions of 3nj coefficients
(asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial,
analytical and numerical tools
New directions in Nielsen-Reidemeister theory
The purpose of this expository paper is to present new directions in the
classical Nielsen-Reidemeister fixed point theory. We describe twisted
Burnside-Frobenius theorem, groups with \emph{property} and a
connection between Nielsen fixed point theory and symplectic Floer homology.Comment: 50 pages, surve
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