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Unformatted text preview: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time Marcus Schaefer Department of Computer Science DePaul University 243 South Wabash Chicago, Illinois 60604, USA [email protected] Eric Sedgwick Department of Computer Science DePaul University 243 South Wabash Chicago, Illinois 60604, USA [email protected] Daniel ˇ Stefankoviˇ c Department of Computer Science University of Rochester Rochester, New York, USA [email protected] December 3, 2007 Abstract Simple curves on surfaces are often represented as sequences of intersections with a trian- gulation. However, there are much more succinct ways of representing simple curves used in topology such as normal coordinates. In these representations, the length of a curve can be exponential in the size of its representation. Nevertheless, we show that the following two basic tasks of computational topology, namely • performing a Dehn-twist of a curve along another curve, and • computing the geometric intersection number of two curves, can be solved in polynomial time even in the succinct normal coordinate representation. These are the first algorithms for these two problems that solve these problems in time polynomial in the succinct representations. As a consequence we can show that a generalized notion of crossing number can be decided in NP, even though the drawings can have exponential complexity. 1 1 Introduction In an earlier paper we started an investigation into algorithms for basic problems of computa- tional topology [SSS02]; we extend this work to deal with crossings of curves in surfaces which has applications to graph drawing. One of the driving problems of computational topology, long before it acquired the name, has been the problem of recognizing the unknot. The story begins with Kneser [Kne30] who, in 1930, introduced a succinct representation for curves and surfaces in which these objects are described by their normal coordinates . This led to the theory of normal surfaces which was used by Haken in 1961 to show that the unknot could be recognized by an algorithm (which, much later, was shown to run in exponential time). Haken’s approach was pushed further by Hass, Lagarias, and Pippenger who exploited the succinctness of the representation to show that the unknot could be recognized in NP [HLP99]. To this end they had to verify in polynomial time that a special type of normal surface was an essential disk, in particular, that is was connected. This result was strengthened by Agol, Hass, and Thurston [AHT02] who showed that the number of connected components of a normal surface can be computed in polynomial time. This immediately implies polynomial time algorithms for checking whether a normal surface is connected, and whether it is orientable....
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- Fall '08
- Computer Science, Computational complexity theory, Manifold