53 research outputs found
Stabilizing Heegaard splittings of toroidal 3-manifolds
Let be a separating incompressible torus in a 3-manifold . Assuming
that a genus Heegaard splitting can be positioned nicely with
respect to (e.g. is strongly irreducible), we obtain an upper
bound on the number of stabilizations required for to become
isotopic to a Heegaard splitting which is an amalgamation along . In
particular, if is a canonical torus in the JSJ decomposition of , then
the number of necessary stabilizations is at most . As a corollary, this
establishes an upper bound on the number of stabilizations required for and any Heegaard splitting obtained by a Dehn twist of
along to become isotopic.Comment: 21 pages, 18 figures. Version for publication. Generalization of the
main theorem and minor changes in style and forma
Degeneration of Heegaard genus, a survey
We survey known (and unknown) results about the behavior of Heegaard genus of
3-manifolds constructed via various gluings. The constructions we consider are
(1) gluing together two 3-manifolds with incompressible boundary, (2) gluing
together the boundary components of surface times I, and (3) gluing a
handlebody to the boundary of a 3-manifold. We detail those cases in which it
is known when the Heegaard genus is less than what is expected after gluing.Comment: This is the version published by Geometry & Topology Monographs on 3
December 200
Lengths of Geodesics on Klein’s Quartic Curve
A well-known and much studied Riemann surface is Klein’s quartic curve. This surface is interesting since it is the smallest complex curve with maximal symmetry. In addition to this high degree of symmetry, Klein’s quartic curve can be tiled by triangles,giving rise to a tiling group generated by reflections. Using the tiling group and the universal cover of the tiling group we are able to compile a list of the lengths of the short,simple,closed geodesics on this surface. In particular,w e are able to determine whether the geodesic loops generated by the tiling are the systoles,i.e.,the shortest closed geodesics
Learning that Matters is Messy: Experiments Revealing Hidden Potential in Higher Education
Why are some learning experiences so profound that they alter our worlds, whereas others don’t end up sticking at all? The author investigates this question in the context of undergraduate education, recounting several educational experiments that highlight subtle but powerful aspects of the student learning experience. By exploring a different approach to teaching a math course, an alternative framework for academic specialization instead of traditional majors, and a radical approach to designing new institutions, an encounter with the hidden, ontological dimension of learning becomes possible. Accessing the ontological experience of the learner opens up new possibilities for meaningful, deep, and transformative learning experiences in higher education
Stabilization, amalgamation, and curves of intersection of Heegaard splittings
We address a special case of the Stabilization Problem for Heegaard
splittings, establishing an upper bound on the number of stabilizations
required to make a Heegaard splitting of a Haken 3-manifold isotopic to an
amalgamation along an essential surface. As a consequence we show that for any
positive integer there are 3-manifolds containing an essential torus and a
Heegaard splitting such that the torus and splitting surface must intersect in
at least simple closed curves. These give the first examples of lower
bounds on the minimum number of curves of intersection between an essential
surface and a Heegaard surface that are greater than one.Comment: Version for publication. To appear in Algebraic and Geometric
Topolog
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