226 research outputs found
The nonorientable four-genus of knots
We develop obstructions to a knot K in the 3-sphere bounding a smooth
punctured Klein bottle in the 4-ball. The simplest of these is based on the
linking form of the 2-fold branched cover of the 3-sphere branched over K.
Stronger obstructions are based on the Ozsvath-Szabo correction term in
Heegaard-Floer homology, along with the G-signature theorem and the
Guillou-Marin generalization of Rokhlin's theorem. We also apply Casson-Gordon
theory to show that for every n greater than one there exists a knot that does
not bound a topologically embedded nonorientable ribbon surface F in the 4-ball
with first Betti number less than n.Comment: 20 pages; expository change
A finite generating set for the level 2 mapping class group of a nonorientable surface
We obtain a finite set of generators for the level 2 mapping class group of a
closed nonorientable surface of genus . This set consists of isotopy
classes of Lickorish's Y-homeomorphisms also called crosscap slides.Comment: 13 pages, 3 figure
The -genus of Kuratowski minors
A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
-genus of a graph is the minimum such that has an
independently even drawing on the orientable surface of genus . An
unpublished result by Robertson and Seymour implies that for every , every
graph of sufficiently large genus contains as a minor a projective
grid or one of the following so-called -Kuratowski graphs: , or
copies of or sharing at most common vertices. We show that
the -genus of graphs in these families is unbounded in ; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its -genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler -genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte
What's wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces
A celebrated result of Mirzakhani states that, if is a finite area
\emph{orientable} hyperbolic surface, then the number of simple closed
geodesics of length less than on is asymptotically equivalent to a
positive constant times , where
denotes the space of measured laminations on . We observed on some explicit
examples that this result does not hold for \emph{nonorientable} hyperbolic
surfaces. The aim of this article is to explain this surprising phenomenon. Let
be a finite area \emph{nonorientable} hyperbolic surface. We show that
the set of measured laminations with a closed one--sided leaf has a peculiar
structure. As a consequence, the action of the mapping class group on the
projective space of measured laminations is not minimal. We determine a partial
classification of its orbit closures, and we deduce that the number of simple
closed geodesics of length less than on is negligible compared to
. We extend this result to general multicurves. Then
we focus on the geometry of the moduli space. We prove that its Teichm\"uller
volume is infinite, and that the Teichm\"uller flow is not ergodic. We also
consider a volume form introduced by Norbury. We show that it is the right
generalization of the Weil--Petersson volume form. The volume of the moduli
space with respect to this volume form is again infinite (as shown by Norbury),
but the subset of hyperbolic surfaces whose one--sided geodesics have length at
least has finite volume. These results suggest that the moduli
space of a nonorientable surface looks like an infinite volume geometrically
finite orbifold. We discuss this analogy and formulate some conjectures
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