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On knot Floer width and Turaev genus
To each knot one can associated its knot Floer homology
, a finitely generated bigraded abelian group. In general, the
nonzero ranks of these homology groups lie on a finite number of slope one
lines with respect to the bigrading. The width of the homology is, in essence,
the largest horizontal distance between two such lines. Also, for each diagram
of there is an associated Turaev surface, and the Turaev genus is the
minimum genus of all Turaev surfaces for . We show that the width of knot
Floer homology is bounded by Turaev genus plus one. Skein relations for genus
of the Turaev surface and width of a complex that generates knot Floer homology
are given.Comment: 15 pages, 15 figure
The two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2
We use geometric techniques to explicitly find the topological structure of
the space of SO(3)-representations of the fundamental group of a closed surface
of genus 2 quotient by the conjugation action by SO(3). There are two
components of the space. We will describe the topology of both components and
describe the corresponding SU(2)-character spaces by parametrizing them by
spherical triangles. There is the sixteen to one branch-covering for each
component, and the branch locus is a union of 2-spheres or 2-tori. Along the
way, we also describe the topology of both spaces. We will later relate this
result to future work into higher-genus cases and the SL(3,R)-representations
Cayley Graphs on Billiard Surfaces
In this article we discuss a connection between two famous constructions in
mathematics: a Cayley graph of a group and a (rational) billiard surface. For
each rational billiard surface, there is a natural way to draw a Cayley graph
of a dihedral group on that surface. Both of these objects have the concept of
"genus" attached to them. For the Cayley graph, the genus is defined to be the
lowest genus amongst all surfaces that the graph can be drawn on without edge
crossings. We prove that the genus of the Cayley graph associated to a billiard
surface arising from a triangular billiard table is always zero or one. One
reason this is interesting is that there exist triangular billiard surfaces of
arbitrarily high genus , so the genus of the associated graph is usually much
lower than the genus of the billiard surface.Comment: 10 pages, 9 figure
Hyperbolic triangular buildings without periodic planes of genus two
We study surface subgroups of groups acting simply transitively on vertex
sets of certain hyperbolic triangular buildings. The study is motivated by
Gromov's famous surface subgroup question: Does every one-ended hyperbolic
group contain a subgroup which is isomorphic to the fundamental group of a
closed surface of genus at least 2? Here we consider surface subgroups of the
23 torsion free groups acting simply transitively on the vertices of hyperbolic
triangular buildings of the smallest non-trivial thickness. These groups gave
the first examples of cocompact lattices acting simply transitively on vertices
of hyperbolic triangular Kac-Moody buildings that are not right-angled. With
the help of computer searches we show, that in most of the cases there are no
periodic apartments invariant under the action of a genus two surface. The
existence of such an action would imply the existence of a surface subgroup,
but it is not known, whether the existence of a surface subgroup implies the
existence of a periodic apartment. These groups are the first candidates for
groups that have no surface subgroups arising from periodic apartments
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