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- Marcus Schaefer, Eric Sedgwick, Daniel Stefankovic
- STOC
- 2002

A <i>string graph</i> is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in <b>NP</b>. The recognition problem was not known to be decidable until very recently, when two independent… (More)

In this paper, we use normal surface theory to study Dehn filling on a knot-manifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knot-manifold that bound normal and almost normal surfaces in a one-vertex triangulation of that knot-manifold. This is combined with existence theorems for normal and almost normal… (More)

- Marcus Schaefer, Eric Sedgwick, Daniel Stefankovic
- CCCG
- 2008

Simple curves on surfaces are often represented as sequences of intersections with a triangulation. However, topologists have much more succinct ways of representing simple curves such as normal coordinates which are exponentially more succinct than intersection sequences. Nevertheless, we show that the following two basic tasks of computational topology,… (More)

To this end let X and Y be 3–manifolds with incompressible boundary homeomorphic to a connected surface F . It is not difficult to show that if HX and HY are Heegaard surfaces in X and Y then we can amalgamate these splittings to obtain a Heegaard surface in X∪F Y with genus equal to g(HX)+g(HY )−g(F) (see, for example, Schultens [14]). Letting g(X), g(Y),… (More)

- Marshall W. Bern, David Eppstein, +19 authors Denis Zorin
- ArXiv
- 1999

Here we present the results of the NSF-funded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. ∗Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc.xerox.com. David Eppstein, Univ. of California, Irvine, Dept.… (More)

- Marcus Schaefer, Eric Sedgwick, Daniel Stefankovic
- COCOON
- 2002

We derive several algorithms for curves and surfaces represented using normal coordinates. The normal coordinate representation is a very succinct representation of curves and surfaces. For embedded curves, for example, its size is logarithmically smaller than a representation by edge intersections in a triangulation. Consequently, fast algorithms for… (More)

- Yo’av Rieck, Eric Sedgwick
- 2002

We show that every thin position for a connected sum of small knots is obtained in an obvious way: place each summand in thin position so that no two summands intersect the same level surface, then connect the lowest minimum of each summand to the highest maximum of the adjacent summand below. See Figure 1. AMS Classification 57M25; 57M27

- Eric Sedgwick
- 2001

Let M be a closed, irreducible, genus two 3–manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M . Then each component manifold Mi of M − F has handle number one, i.e. admits a Heegaard splitting obtained by attaching a single 1–handle to one or two components of ∂Mi. This result also holds for… (More)

- Eric Sedgwick
- 2004

Suppose that a three-manifold M contains infinitely many distinct strongly irreducible Heegaard splittings H + nK, obtained by Haken summing the surface H with n copies of the surface K. We show that K is incompressible. All known examples, of manifolds containing infinitely many irreducible Heegaard splittings, are of this form. We also give new examples… (More)

- YOAV MORIAH, Eric Sedgwick, Eric Sedgwick
- 2007

We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilized Heegaard surface destabilizes, and… (More)