147 research outputs found
Partial duals of plane graphs, separability and the graphs of knots
There is a well-known way to describe a link diagram as a (signed) plane
graph, called its Tait graph. This concept was recently extended, providing a
way to associate a set of embedded graphs (or ribbon graphs) to a link diagram.
While every plane graph arises as a Tait graph of a unique link diagram, not
every embedded graph represents a link diagram. Furthermore, although a Tait
graph describes a unique link diagram, the same embedded graph can represent
many different link diagrams. One is then led to ask which embedded graphs
represent link diagrams, and how link diagrams presented by the same embedded
graphs are related to one another. Here we answer these questions by
characterizing the class of embedded graphs that represent link diagrams, and
then using this characterization to find a move that relates all of the link
diagrams that are presented by the same set of embedded graphs.Comment: v2: major change
Angled decompositions of arborescent link complements
This paper describes a way to subdivide a 3-manifold into angled blocks,
namely polyhedral pieces that need not be simply connected. When the individual
blocks carry dihedral angles that fit together in a consistent fashion, we
prove that a manifold constructed from these blocks must be hyperbolic. The
main application is a new proof of a classical, unpublished theorem of Bonahon
and Siebenmann: that all arborescent links, except for three simple families of
exceptions, have hyperbolic complements.Comment: 42 pages, 23 figures. Slightly expanded exposition and reference
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
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
Fiber detection for state surfaces
Every Kauffman state \sigma of a link diagram D(K) naturally defines a state
surface S_\sigma whose boundary is K. For a homogeneous state \sigma, we show
that K is a fibered link with fiber surface S_\sigma if and only if an
associated graph G'_\sigma is a tree. As a corollary, it follows that for an
adequate knot or link, the second and next-to-last coefficients of the Jones
polynomial are obstructions to certain state surfaces being fibers for K.
This provides a dramatically simpler proof of a theorem from
[arXiv:1108.3370].Comment: 6 pages, 5 figures. v2 features minor revisions. To appear in
Algebraic & Geometric Topolog
Spectrally Similar Incommensurable 3-Manifolds
Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds Nn and Nµn whose volume is approximately n and whose length spectra agree up to length n.
Both Nn and Nµn are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that these manifolds have a very thick collar about an essential surface. The two gluing maps differ by a hyper-elliptic involution along this surface. Our proof also involves a new commensurability criterion based on pairs of pants
- …