414 research outputs found
Geodesics and compression bodies
We consider hyperbolic structures on the compression body C with genus 2
positive boundary and genus 1 negative boundary. Note that C deformation
retracts to the union of the torus boundary and a single arc with its endpoints
on the torus. We call this arc the core tunnel of C. We conjecture that, in any
geometrically finite structure on C, the core tunnel is isotopic to a geodesic.
By considering Ford domains, we show this conjecture holds for many
geometrically finite structures. Additionally, we give an algorithm to compute
the Ford domain of such a manifold, and a procedure which has been implemented
to visualize many of these Ford domains. Our computer implementation gives
further evidence for the conjecture.Comment: 31 pages, 11 figures. V2 contains minor changes. To appear in
Experimental Mathematic
Links with no exceptional surgeries
We show that if a knot admits a prime, twist-reduced diagram with at least 4
twist regions and at least 6 crossings per twist region, then every non-trivial
Dehn filling of that knot is hyperbolike. A similar statement holds for links.
We prove this using two arguments, one geometric and one combinatorial. The
combinatorial argument further implies that every link with at least 2 twist
regions and at least 6 crossings per twist region is hyperbolic and gives a
lower bound for the genus of a link.Comment: 28 pages, 15 figures. Minor rewording and organizational changes;
also added theorem giving a lower bound on the genus of these link
Recommended from our members
Are personalities genetically determined? Inferences from subsocial spiders.
BACKGROUND:Recent research has revealed that polymorphic behavioral strategies shape intra-and interspecific interactions and contribute to fitness in many animal species. A better understanding of the proximate mechanisms underlying these behavioral syndromes will enhance our grasp this phenomenon. Spiders in the genus Anelosimus exhibit inter-individual behavioral variation on several axes: individuals have consistent responses to stimuli (e.g. bold vs. shy individuals) and they are subsocial (exhibiting extended maternal care and sibling cooperation) across most of their range, but they sometimes form permanent social groups in northern temperate regions. Here, we seek genetic variants associated with boldness and with social structure in a socially polymorphic population of the spider Anelosimus studiosus. We also develop preliminary genomic resources, including a genome assembly and linkage map, that support this and future genomic research on this group. RESULTS:Remarkably, we identify a small genomic scaffold (~ 1200 bp) that harbors seven single nucleotide polymorphisms (SNPs) associated with boldness. Moreover, heterozygotes are less common than expected based on Hardy-Weinberg equilibrium, suggesting that either assortative mating or selection against heterozygotes may be occurring in this system. We find no loci significantly associated with social organization. Our draft genome assembly allows us to localize SNPs of interest in this study and to carry out genetic comparisons with other published genomes, although it remains highly fragmented. CONCLUSIONS:By identifying a locus associated with a well-studied animal personality trait, this study opens up avenues for future research to link behavioral studies of animal personality with genotype and fitness
Treewidth, crushing, and hyperbolic volume
We prove that there exists a universal constant such that any closed
hyperbolic 3-manifold admits a triangulation of treewidth at most times its
volume. The converse is not true: we show there exists a sequence of hyperbolic
3-manifolds of bounded treewidth but volume approaching infinity. Along the
way, we prove that crushing a normal surface in a triangulation does not
increase the carving-width, and hence crushing any number of normal surfaces in
a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former
argument (in V1) used a construction that relied on a wrong theorem. Section
5.1 has also been adjusted to the new construction. Various other arguments
have been clarifie
Explicit Dehn filling and Heegaard splittings
We prove an explicit, quantitative criterion that ensures the Heegaard
surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic
manifold X, and a Dehn filling whose meridian and longitude curves are longer
than 2pi(2g-1), we show that every genus g Heegaard splitting of the filled
manifold is isotopic to a splitting of the original manifold X. The analogous
statement holds for fillings of multiple boundary tori. This gives an effective
version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.Comment: 17 pages. v3 contains minor revisions and cleaner arguments,
incorporating referee comments. To appear in Communications in Analysis and
Geometr
- …