5,504 research outputs found
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
Chord Diagrams and Gauss Codes for Graphs
Chord diagrams on circles and their intersection graphs (also known as circle
graphs) have been intensively studied, and have many applications to the study
of knots and knot invariants, among others. However, chord diagrams on more
general graphs have not been studied, and are potentially equally valuable in
the study of spatial graphs. We will define chord diagrams for planar
embeddings of planar graphs and their intersection graphs, and prove some basic
results. Then, as an application, we will introduce Gauss codes for immersions
of graphs in the plane and give algorithms to determine whether a particular
crossing sequence is realizable as the Gauss code of an immersed graph.Comment: 20 pages, many figures. This version has been substantially
rewritten, and the results are stronge
The decay of the X(3872) into \chi_{cJ} and the Operator Product Expansion in XEFT
XEFT is a low energy effective theory for the X(3872) that can be used to
systematically analyze the decay and production of the X(3872) meson, assuming
that it is a weakly bound state of charmed mesons. In a previous paper, we
calculated the decays of X(3872) into \chi_{cJ} plus pions using a two-step
procedure in which Heavy Hadron Chiral Perturbation Theory (HH\chiPT)
amplitudes are matched onto XEFT operators and then X(3872) decay rates are
then calculated using these operators. The procedure leads to IR divergences in
the three-body decay X(3872) \to \chi_{cJ} \pi \pi when virtual D mesons can go
on-shell in tree level HH\chiPT diagrams. In previous work, we regulated these
IR divergences with the width. In this work, we carefully analyze
X(3872) \to \chi_{cJ} \pi^0 and X(3872) \to \chi_{cJ} \pi \pi using the
operator product expansion (OPE) in XEFT. Forward scattering amplitudes in
HH\chiPT are matched onto local operators in XEFT, the imaginary parts of which
are responsible for the decay of the X(3872). Here we show that the IR
divergences are regulated by the binding momentum of the X(3872) rather than
the width of the D^{*0} meson. In the OPE, these IR divergences cancel in the
calculation of the matching coefficients so the correct predictions for the
X(3872) \to \chi_{c1} \pi \pi do not receive enhancements due to the width of
the D^{*0}. We give updated predictions for the decay X(3872) \to \chi_{c1} \pi
\pi at leading order in XEFT.Comment: 20 pages, 10 figure
An Introduction to Virtual Spatial Graph Theory
Two natural generalizations of knot theory are the study of spatial graphs
and virtual knots. Our goal is to unify these two approaches into the study of
virtual spatial graphs. This paper is a survey, and does not contain any new
results. We state the definitions, provide some examples, and survey the known
results. We hope that this paper will help lead to rapid development of the
area.Comment: 9 pages, 7 figures, presented at the International Workshop on Knot
Theory for Scientific Objects at Osaka City University, March 200
Homotopy on spatial graphs and the Sato-Levine invariant
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs
which are generalizations of Milnor's link-homotopy. We introduce some edge
(resp. vertex)-homotopy invariants of spatial graphs by applying the
Sato-Levine invariant for the 2-component constituent algebraically split links
and show examples of non-splittable spatial graphs up to edge (resp.
vertex)-homotopy, all of whose constituent links are link-homotopically
trivial.Comment: 17 pages,16 figure
Intrinsically linked graphs and even linking number
We study intrinsically linked graphs where we require that every embedding of
the graph contains not just a non-split link, but a link that satisfies some
additional property. Examples of properties we address in this paper are: a two
component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where
all linking numbers are even, or an n-component link with components L, A_i
where lk(L,A_i) = 3k, k not 0. Links with other properties are considered as
well. For a given property, we prove that every embedding of a certain complete
graph contains a link with that property. The size of the complete graph is
determined by the property in question.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-55.abs.htm
Some Considerations of the Goal Setting and Planning Processes in Public Higher Education
Too frequently the question of institutional purpose in higher education is unaddressed despite common agreement that it constitutes the critical core of organizational life. Resultantly, universities pursue complex programmes with little internal or external consensus on fundamental goals. This paper examines some important decision making areas in higher education and isolates a number of critical institutional variables which determine the nature of planning.La question d'objectif institutionnel est trop souvent escamotée malgré qu'elle constitue, selon l'accord commun, le noyau critique de la vie organisationnelle. Il en résulte que les universités poursuivent des programmes complexes sans trop chercher de consensus ni interne ni externe sur les objectifs fondamentaux. Cette étude fait l'analyse de quelques domaines importants de prise de décision dans l'enseignement supérieur et isole un certain nombre de variables institutionnels critiques qui déterminent l'orientation de la planification
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