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Transactions of the American Mathematical Society

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The connectivity of multicurves determined by integral weight train tracks


Authors: Andrew Haas and Perry Susskind
Journal: Trans. Amer. Math. Soc. 329 (1992), 637-652
MSC: Primary 57N05
DOI: https://doi.org/10.1090/S0002-9947-1992-1028309-1
MathSciNet review: 1028309
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Abstract: An integral weighted train track on a surface determines the isotopy class of an embedded closed $ 1$-manifold. We are interested in the connectivity of the resulting $ 1$-manifold. In general there is an algorithm for determining connectivity, and in the simplest case of a $ 2$-parameter train track on a surface of genus one there is an explicit formula. We derive a formula for the connectivity of the closed $ 1$-manifold determined by a $ 4$-parameter train track on a surface of genus two which is computable in polynomial time. We also give necessary and sufficient conditions on the parameters for the resulting $ 1$-manifold to be connected.


References [Enhancements On Off] (What's this?)

  • [1] A. Haas and P. Susskind, The geometry of the hyperelliptic involution in genus two, Proc. Amer. Math. Soc. 105 (1989), 159-165. MR 930247 (89e:30078)
  • [2] R. C. Penner, An introduction to train tracks, Low-dimensional Topology and Kleinian Groups, Warwick and Durham, 1984, (Epstein, ed.), LMS Lecture Notes Ser. 112, Cambridge Univ. Press, Cambridge and New York, 1986, pp. 77-90. MR 903859 (89e:57005)
  • [3] J. Harer and R. C. Penner, Combinatories of train tracks, unpublished manuscript.
  • [4] W. Thurston, The geometry and topology of $ 3$-manifolds, Princeton Univ. Notes.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1028309-1
Keywords: Train track, surface topology, multiple curves
Article copyright: © Copyright 1992 American Mathematical Society

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