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On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces

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References

  1. Birman, J.S., González-Acuña, F., Montesinos, J.M.: Heegard splittings of prime 3-manifolds are not unique. Mich. Math. J.23, 97–103 (1976)

    Google Scholar 

  2. Boileau, M., Otal, J.P.: Groupes des difféotopies de certaines varietés de Seifert. C. R. Acad. Sci. Paris303-I, 19–22 (1986)

    Google Scholar 

  3. Boileau, M., Rost, M., Zieschang, H.: Décompositions de Heegaard des éxtérieurs des noeuds toriques et des variétés de Seifert associées. C.R. Acad. Sci. Paris302-I, 661–664 (1986)

    Google Scholar 

  4. Boileau, M., Zieschang, H.: Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math.76, 455–468 (1984)

    Google Scholar 

  5. Burde, G., Zieschang, H.: Eine Kennzeichnung der Torusknoten. Math. Ann.167, 169–176 (1966)

    Google Scholar 

  6. Burde, G., Zieschang, H.: Knots. Berlin: de Gruyter 1985

    Google Scholar 

  7. Clark, B.: The Heegaard genus of manifolds obtained by surgery on links and knots. Int. J. Math. Sci.3, 583–589 (1980)

    Google Scholar 

  8. Collins, D.J.: Presentations of the amalgamated free product of two infinite cycles. Math. Ann.237, 233–241 (1978)

    Google Scholar 

  9. Floyd, W.: Hyperbolic manifolds obtained by adding a 2-handle to a 3-manifold with compressible boundary. Preprint

  10. Funcke, K.: Gegenbeispiele zu einer Vermutung von Magnus. Math. Z.141, 205–217 (1975)

    Google Scholar 

  11. Johannson, K.: On surfaces in one-relator 3-manifolds. Preprint

  12. Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  13. Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory: presentations of groups in terms of generators and relations. New York: Wiley 1966

    Google Scholar 

  14. Moriah, Y.: On Heegaard splittings and group presentations. (Preprint and Ph.D. thesis, University of Texas, Austin 1986)

    Google Scholar 

  15. Moriah, Y.: Heegaard splittings of Seifert fibered spaces. Preprint

  16. Nielsen, J.: Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Math. Ann.78, 385–397 (1918)

    Google Scholar 

  17. Orlik, P., Vogt, E., Zieschang, H.: Zur Topologie gefaserter dreidimensionaler Mannigfaltigkeiten. Topology6, 49–64 (1967)

    Google Scholar 

  18. Osborne, R. P., Zieschang, H.: Primitives in the free group on two generators. Invent. Math.63, 17–24 (1981)

    Google Scholar 

  19. Reidemeister, K.: Über Heegaard-Diagramme. Abh. Math. Semin. Hamb.25, 140–145 (1962)

    Google Scholar 

  20. Scharlemann, M.: Tunnel number one knots satisfy the Poincaré conjecture. Topology Appl.18, 235–258 (1984)

    Google Scholar 

  21. Scharlemann, M.: The Thurston norm and 2-handle addition. Preprint

  22. Schreier, O.: Über die GruppenA a B b=1. Abh. Math. Semin. Univ. Hamb.3, 167–169 (1924)

    Google Scholar 

  23. Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc.15, 401–487 (1983)

    Google Scholar 

  24. Seifert, H.: Topologie dreidimensionaler gefaserter Räume. Acta. Math.60, 147–238 (1933)

    Google Scholar 

  25. Waldhausen, F.: Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten. Topology6, 231–250 (1962)

    Google Scholar 

  26. Waldhausen, F.: Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I.. Invent. Math.3, 308–333 (1967); II, Invent. Math.4, 87–117 (1967)

    Google Scholar 

  27. Zieschang, H.: Über einfache Kurven auf Vollbrezeln. Abh. Math. Semin. Univ. Hamb.25, 231–250 (1962)

    Google Scholar 

  28. Zieschang, H.: Über einfache Kurvensysteme auf einer Vollbrezel vom Geschlecht 2. Abh. Math. Semin. Univ. Hamb.26, 237–247 (1963)

    Google Scholar 

  29. Zieschang, H.: Classification of simple systems of paths on a solid pretzel of genus 2. Sov. Math.4, 1460–1463 (1963) [Transl. Dokl. Acad. Sci. USSR152, 841–844 (1963)]

    Google Scholar 

  30. Zieschang, H.: Alternierende Produkte in freien Gruppen. Abh. Math. Semin. Univ. Hamb.27, 13–31 (1964)

    Google Scholar 

  31. Zieschang, H.: Generators of the free product with amalgamation of two infinite cyclic groups. Math. Ann.277, 195–221 (1977)

    Google Scholar 

  32. Zieschang, H., Vogt, E., Coldewey, H.-D.: Surfaces and planar discontinuous groups. Lecture Notes in Math. Vol. 835. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

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Boileau, M., Rost, M. & Zieschang, H. On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces. Math. Ann. 279, 553–581 (1988). https://doi.org/10.1007/BF01456287

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