Abstract
Consider a general position immersion of a circle into the 2-sphere. Suppose the immersion has an even number of double points. Then there is a proper immersion of the 2-disk that has the given curve as its boundary. Of all such extentions there is one with a minimum number of triple points. This minimum is obtained algorithmically in terms of a number that is associated to the double point set.
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References
Bailey, K. D.,Extending Closed Plane Curves to Immersions of the Disk with n Handles, Trans. AMS206 (1975), 1–24.
Banchoff, T. F.,Triple Points and Surgery of Immersed Surfaces, Proc. AMS46 No. 3 (Dec. 1974), 403–413.
Bredon, Glen E. andWood, John W.,Non-orientable Surfaces in Orientable 3-manifolds, Inventiones Math.7 (1969), 83–110.
Carter, J. Scott,Surgery on Codimension One Immersions in (n+1)-space: Removing n-tuple Points, Trans. AMS298 No. 1 (Nov. 1986), 83–102.
Carter, J. Scott,On Generalizing Boy's Surface: Constructing a Generator of the Third Stable Stem, Trans. AMS298, No. 1 (Nov. 1986), 103–122.
Carter, J. Scott,A Further Generalization of Boy's Surface, Houston Journal of Mathematics12, No. 1 (1986), 11–31.
Carter, J. Scott,Surgery on the Equatorial Immersion I, Illinois Journal of Mathematics, Vol.34, No. 4 (1988), 704–715.
Carter, J. Scott,Surgering the Equatorial Immersion in Low Dimensions, Difierential Topology Proceedings, Siegen 1987, ed. Ulrich Koschorke, LNM 1350.
Carter, J. Scott,Immersed Codimension One Projective Spaces in Spherical Space Forms, Proc. of the AMS105 No. 1 (Jan. 1989), 254–257.
Carter, J. Scott,Classifying Immersed Curves, Proc. of the AMS111, No. 1 (Jan. 1991), 281–287.
Carter, J. Scott,Extending Immersions of Curves to Properly Immersed Surfaces, Topol. and its Appl.40, No. 3 (Aug. 1991), 287–306.
Dowker, C. H., andThistlethwaite, M. W.,Classifications of Knot Porjections, Topology and its Applications16 (1983), 19–31.
Fenn, Roger, Techniques of Geometric Topology, London Math. Society Leuture Note Series:57, Cambridge Univ. Press (1983), 71–87.
Francis, George,Extensions to the Disk of Properly Nested Plane Immersions, Michigan Math. J.17 (1970), 377–383.
Gauss, C. F., Werke VIII, pages 271–286.
Haas andHughes,Immersions of Surfaces in 3-manifolds, Topology24, No. 1 (1985), 97–112.
Lovasz, L. andMarx, M. L.,A Forbidden Substructure Characterization of Gauss Codes Acta Sci. Math.38 (1976), 115–119.
Poenaru, V.,Extensions des immersions en codimension 1 (d'apres Blank), Seminar Bourbaki, 1967/68, Expose 342, Benjamin, New York (1969).
Read, R. C. andRosensteihl, P.,On the Gauss Crossing Problem, Coloquia Mathematica Societatis Janos Bolyai, 18 Combinatorics, Keszthely (Hungary) (1976), 843–877.
Trace, Bruce,A General Position Theorem for Surfaces in 4-space, AMS Comtemporary Mathematics Series v.44 (1985), 123–137.
Whitney, H.,On Regular Closed Curves in the Plane, Compositio Math.4 (1937), 276–284.
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Carter, J.S. Extending immersed circles in the sphere to immersed disks in the ball. Commentarii Mathematici Helvetici 67, 337–348 (1992). https://doi.org/10.1007/BF02566506
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DOI: https://doi.org/10.1007/BF02566506