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Unformatted text preview: Computational Topology (Jeff Erickson) Normal Surfaces and Knots In science there are no ’depths’; there is surface everywhere: all experience forms a complex network, which cannot always be surveyed and can often be grasped only in parts. — Rudolf Carnap, Hans Hahn, and Otto Neurath, The Scientific Conception of the World: The Vienna Circle (1929) The perfidious lemma of Dehn Was every topologist’s bane ’Til Christos Papa- kyriakopou- los proved it without any strain. — John Milnor (c. 1956) 14 Normal Surfaces and Knots 14.1 Definitions Following Kneser [ 8 ] and Haken [ 4 ] , we say that an embedded surface S in a 3-manifold M is normal with respect to a fixed triangulation T of M if (1) all intersections between S and T are transverse, and (2) the intersection of S with each tetrahedron in T is a set of disjoint elementary disks. An elementary disk in a tetrahedron intersects either three or four edges of the tetrahedron, each exactly once; in other words, it looks just like the intersection of a Euclidean plane with a Euclidean tetrahedron. Each tetrahedron can support exactly seven different types of elementary disks: four triangles (each cutting off one vertex of the tetrahedron) and three quadrilaterals (each separating two vertices from two others). Because S is embedded, at most one type of quadrilateral can appear in any tetrahedron. Five of the seven types of elementary disks inside a tetrahedron For any normal surface S and any tetrahedron pqrs , let S p | qrs denote the number of elementary triangles that separate vertex w from the other vertices, and let S pq | rs denote the number of elementary quadrilaterals separating p and q from r and s . The normal surface S can be described by a vector 〈 S 〉 of 7 t normal coordinates where t is the number of tetrahedra in T ; these include 4 t triangle coordinates S w | x yz and 3 t quadrilateral coordinates S wx | yz . Haken proved that a vector 〈 S 〉 ∈ N 7 t is a normal coordiante vector if and only if it satisfies two constraints. First, different tetrahedra must agree on the number of intersections with each edge, as well as the number of each type of elementary segment on each triangle. It suffices for 〈 S 〉 to satisfy the following consistency constraint ; if pqrz and apqr are two tetrahedra sharing a common face pqr , then S p | aqr + S ap | qr = S p | qrz + S pz | qr . Second, within each tetrahedron, at most one of of the quadrilateral coordinates can be non-zero. The normal-curve algorithms of Schaefer, Sedgwick, and Štefankoviˇ c [ 11 ] are easily adapted to similar problems on normal surfaces. 1 Computational Topology (Jeff Erickson) Normal Surfaces and Knots Theorem 14.1. Given the normal coordinate vector 〈 S 〉 of a normal surface S , we can compute the number of components of S and the Euler characteristic of each component in polynomial time....
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- Fall '08