105 research outputs found
Orbifolds and commensurability
These are notes based on a series of talks that the author gave at the
"Interactions between hyperbolic geometry and quantum groups" conference held
at Columbia University in June of 2009.Comment: 11 page
Great circle links and virtually fibered knots
We show that all two-bridge knot and link complements are virtually fibered.
We also show that spherical Montesinos knot and link complements are virtually
fibered. This is accomplished by showing that such knot complements are
finitely covered by great circle link complements.Comment: Minor changes. To appear in Topology. 12 pages, 7 figure
The big Dehn surgery graph and the link of S^3
In a talk at the Cornell Topology Festival in 2005, W. Thurston discussed a
graph which we call "The Big Dehn Surgery Graph", B. Here we explore this
graph, particularly the link of S^3, and prove facts about the geometry and
topology of B. We also investigate some interesting subgraphs and pose what we
believe are important questions about B.Comment: 15 pages, 4 figures, 4 ancillary files. Reorganized and shortened
from previous versions, while correcting one error in the proof of Theorem
5.4. Also, ancillary files detailing our computations with the computer
program ORB have been provide
Great circle links in the three-sphere
We investigate great circle links in the three-sphere, the class of links
where each component is a great circle. Using the geometry of their
complements, we classify such links up to five components. For any two-bridge
knot complement, there is a finite cover that is the complement of a link of
great circles in . We show that for many two-bridge knots, this cover
contains a closed incompressible surface. Infinitely many fillings of the
two-bridge knot lift to fillings of great circle link where the
incompressibility of this surface is preserved. Using this, we show that
infinitely many fillings of an infinite class of two-bridge knot complements
are virtually Haken.Comment: Dissertation for degree awarded June 17, 2003 from UC Davis, 50
pages, 18 figure
The automorphism group of the free group of rank two is a CAT(0) group
We prove that the automorphism group of the braid group on four strands acts
faithfully and geometrically on a CAT(0) 2-complex. This implies that the
automorphism group of the free group of rank two acts faithfully and
geometrically on a CAT(0) 2-complex, in contrast to the situation for rank
three and above.Comment: 7 pages, 2 figures. The manuscript has been modified in minor ways in
accordance with a referee's recommendations, and a misattribution of the
result "Aut F_2 is biautomatic" has been correcte
Double bubbles in the 3-torus
We present a conjecture, based on computational results, on the area
minimizing way to enclose and separate two arbitrary volumes in the flat cubic
3-torus. For comparable small volumes, we prove that an area minimizing double
bubble in the 3-torus is the standard double bubble from R^3.Comment: 13 pages, 4 figures. Prepared on behalf of the participants in the
Clay Mathematics Institute Summer School on the Global Theory of Minimal
Surfaces, held at the Mathematical Sciences Research Institute in Berkeley,
California, Summer 200
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