Stabilizing Heegaard splittings of toroidal 3-manifolds

Let $T$ be a separating incompressible torus in a 3-manifold $M$. Assuming that a genus $g$ Heegaard splitting $V \cup_S W$ can be positioned nicely with respect to $T$ (e.g. $V \cup_S W$ is strongly irreducible), we obtain an upper bound on the number of stabilizations required for $V \cup_S W$ to become isotopic to a Heegaard splitting which is an amalgamation along $T$. In particular, if $T$ is a canonical torus in the JSJ decomposition of $M$, then the number of necessary stabilizations is at most $4g-4$. As a corollary, this establishes an upper bound on the number of stabilizations required for $V \cup_S W$ and any Heegaard splitting obtained by a Dehn twist of $V \cup_S W$ along $T$ 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

Transportation in McMinnville 100 Years Ago

Our project presents a historical view of transportation and its development in McMinnville in the 1910s, especially in 1912. McMinnville was incorporated as a town in 1876 and became a city in 1882. At the time, people still frequently rode horses (including stagecoaches), and the Yamhill River provided an early artery for pioneer travel and commerce. Then rails expanded in Oregon in the late 1800s and became a popular mode of transportation in the early 20th century. By the 1910s, horses, steamboats, rails, and cars were four major modes of transportation in McMinnville

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 $n$ there are 3-manifolds containing an essential torus and a Heegaard splitting such that the torus and splitting surface must intersect in at least $n$ 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

Response and optimization of an isolation system with relaxation type damping

Response and optimization of isolation system with relaxation type dampin