Eric Sedgwick

Recognizing string graphs in NP

A string graph 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 NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph [18, 20]. These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NP-complete, since Kratochvíl [12] showed that the recognition problem is NP-hard. The result has consequences for the computational complexity of problems in graph drawing, and topological inference.

Decision problems in the space of Dehn fillings

Normal surface theory is used to study Dehn fillings of a knot-manifold. We use that any triangulation of a knot-manifold may be modified to a triangulation having just one vertex in the boundary. In this situation, it is shown that there is a finite computable set of slopes on the boundary of the knot-manifold, which come from boundary slopes of normal or almost normal surfaces. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely those manifolds obtained by Dehn filling of a given knot-manifold that: (1) are reducible, (2) contain two-sided incompressible surfaces, (3) are Haken, (4) fiber over S1, (5) are the 3-sphere, and (6) are a lens space. Each of these algorithms is a finite computation. Moreover, in the case of essential surfaces, we show that the topology of each filled manifold is strongly reflected in the combinatorial properties of a triangulation of the knot-manifold with just one vertex in the boundary. If a filled manifold contains an essential surface then the knot-manifold contains an essential normal vertex solution which caps off to an essential surface of the same type in the filled manifold. (Normal vertex solutions are the premier class of normal surface and are computable.)

Thin position for a connected sum of small knots

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 Classication 57M25; 57M27

Persistence of Heegaard structures under Dehn filling

It is well known that a Heegaard surface may destabilize after Dehn filling, reducing the genus by one or more. This phenomenon is classified according to whether or not the core of the attached solid torus is isotopic into the destabilized surface. When it is, the destabilized surface will be a Heegaard surface for infinitely many fillings, arranged along a destabilization line in the Dehn surgery space. Here we demonstrate that a destabilization line corresponds to a slope bounding an essential surface. Such slopes are known to be finite in number and therefore so is the number of destabilization lines. We apply this result to study Heegaard genus. In particular we prove, using purely topological techniques, that if X is any a-cylindrical manifold, then there are an infinite number of Dehn fillings on X which produce a manifold of the same genus as X.

Embeddability in the 3-sphere is decidable

We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in R3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S3, then there is also an embedding in which X has a short meridian, i.e., an essential curve in the boundary of X bounding a disk in S3 \ X with length bounded by a computable function of the number of tetrahedra of X.

Algorithms for Normal Curves and Surfaces

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 normal representations can be exponentially faster than algorithms working on the edge intersection representation. Normal representations have been essential in establishing bounds on the complexity of recognizing the unknot [Hak61, HLP99, AHT02], and string graphs [SS?02]. In this paper we present efficient algorithms for counting the number of connected components of curves and surfaces, deciding whether two curves are isotopic, and computing the algebraic intersection numbers of two curves. Our main tool are equations over monoids, also known as word equations.

Genus two 3{manifolds are built from handle number one pieces

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 M i of M - F has handle numberat most one, i.e. admits a Heegaard splitting obtained by attaching a single 1-handle to one or two components of ∂M i . This result also holds for a decomposition of M along a maximal collection of incompressible tori.

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

Genus characterizes the complexity of graph problems: some tight results

We study the fixed-parameter tractability, subexponential time computability, and approximability of the well-known NP-hard problems: Independent Set, Vertex Cover, and Dominating Set. We derive tight results and show that the computational complexity of these problems, with respect to the above complexity measures, is dependent on the genus of the underlying graph. For instance, we show that, under the widely-believed complexity assumption W[1] ≠ FPT, INDEPENDENT SET on graphs of genus bounded by g1(n) is fixed parameter tractable if and only if g1(n) = o(n2), and DOMINATING SET on graphs of genus bounded by g2(n) is fixed parameter tractable if and only if g2(n) = no(1). Under the assumption that not all SNP problems are solvable in subexponential time, we show that the above three problems on graphs of genus bounded by g3(n) are solvable in subexponential time if and only if g3(n) = o(n).

Spiraling and Folding: The Word View

We show that for every n there are two simple curves on the torus intersecting at least n times without the two curves folding or spiraling with respect to each other. On the other hand, two simple curves in a punctured plane that intersect at least n times (and do not create any empty bigons) must either form a spiral of depth d or a fold of width cn/(d+1)−1, where c only depends on the number of punctures in the plane. The construction of the two curves on the torus involves train tracks and word equations, and the verification that the two curves do not spiral leads us to an infinite binary word based on the golden ratio which does not contain any square word ww for which |w| is even.