References
Birman, J.S.: Braids, links, and mapping class groups. Ann. Math. Stud.82, (1974)
Birman, J.S., Hilden, H.M.: On the mapping class groups of closed surfaces as covering spaces. In: Advances in the theory of Riemann surfaces. Ann. Math. Stud.66, 81–115 (1971)
Bleiler, A.S., Moriah, Y.: Heegaard splittings and branched coverings of B3. Math. Ann.281, 531–543 (1988)
Boileau, M., Collins, D.J., Zieschang, H.: Scindements de Heegaard des petites variétés de Seifert. C.R. Acad. Sci. Paris303-I, 557–560 (1987)
Boileau, M., Collins, D.J., Zieschang, H.: Genus 2 Heegaard decompositions of small Seifert manifolds. Preprint
Boileau, M., Rost, M., Zieschang, H.: On Heegaard decompositions of torus exteriors and related Seifert fibred spaces. Math. Ann.279, 553–581 (1988)
Bonahon, F., Siebenmann, L.: Geometric splittings of knots, and Conway's algebraic knots. Preprint
Burde, G., Zieschang, H.: Knots. De Gruyter Studies in Mathematics 5. Berlin New York: de Gruyter 1985
Boyle, J.: Classifying 1-handles attached to knotted surfaces. Trans. Am. Math. Soc.306, 475–487 (1988)
Collins, D.J.: Presentations of the amalgamated free products of two infinite cycles. Math. Ann.237, 233–241 (1978)
Culler, M., Gordon, C.McA., Lueke, J., Shalen, P.B.: Dehn surgery on knots. Ann. Math.125, 237–300 (1987)
Epstein, D.B.A.: Curves on 2-manifolds and isotopies. Acta Math.115, 83–107 (1966)
Fukuhara, S., Kanno, J.: Extended Alexander matrices of 3-manifolds. I.. Tokyo J. Math.8, 107–120 (1985)
Funke, K.: Nicht freie äquivalente Darstellungen mit einer definierenden Relation. Math. Z.141, 205–217 (1975)
Gabai, D.: Detecting fibred links inS 3. Comment. Math. Helv.61, 519–555 (1986)
Gordon, C.McA.: On primitive sets of loops in the boundary of a handlebody. Topology Appl.27, 285–299 (1987)
Gordon, C.McA., Litherland, R.A., Murasugi, K.: Signature of covering links. Can. J. Math.33, 381–394 (1981)
Handel, M.: New proofs of some results of Nielsen. Adv. Math.56, 173–191 (1985)
Hartley, R., Kawauchi, A.: Polynomials of amphicheiral knots. Math. Ann.243, 63–70 (1979)
Jaco, W.: Lectures on three manifolds topology. CBMS Reg. Conf. Ser. Math.43 (1980)
Jaco, W., Shalen, P.B.: Seifert fibred space in 3-manifolds. Mem. Am. Math. Soc.220 (1979)
Johannson, K.: Homotopy equivalences of 3-manifolds with boundaries. Lect. Notes in Math.761. Berlin Heidelberg New York: Springer 1979
Kobayashi, T.: Structures of the Haken manifolds with Heegaard splittings of genus two. Osaka J. Math.21, 437–455 (1984)
Kobayashi, T.: Structures of full Haken manifolds. Osaka J. Math.24, 173–215 (1987)
Kobayashi, T.: A criterion for detecting inequivalent tunnels for a knot. To appear in Math. Proc. Camb. Phil. Soc.107, 483–491 (1990)
Lustig, M., Moriah, Y.: Nielsen equivalence in Fuchsian groups and Seifert fibred spaces. Preprint
Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. Ergeb. Math. Grenzgeb. 89. Berlin Heidelberg New York: Springer 1977
Moriah, Y.: On Heegaard splittings and group presentations. Thesis, Univ. Texas, 1986
Moriah, Y.: Heegaard splittings of Seifert fibred spaces. Invent. Math.91, 465–481 (1988)
Morimoto, K.: On minimum genus Heegaard splittings of some orientable closed 3-manifolds. Tokyo J. Math.12, 321–355 (1989)
Moser, L.: Elementary surgery along a torus knot. Pacific J. Math.38, 737–745 (1971)
Murasugi, K.: Seifert fibre spaces and braid groups. Proc. Lond. Math. Soc.44, 71–84 (1982)
Norwood, F.: Every two generator knot is prime. Proc. Am. Math. Soc.86, 143–147 (1982)
Rolfsen, D.: Knots and links. Math. Lect. Note Series 7. Publish or Perish Inc., 1976
Orlik, P.: Seifert Manifolds. Lecture Notes in Mathematics 291, Berlin Heidelberg New York: Springer 1972
Sakuma, M.: On strongly invertible knots. In: Algebraic and topological theories. Proc. Miyata Memorial Conf. Kinokuniya Co. Ltd., 1985
Sakuma, M.: Realization of the symmetry groups of links. In: Transformation groups. Lect. Notes in Math. 1375, pp. 291–306. Berlin Heidelberg New York: Springer 1989
Scharlemann, M.: Tunnel number one knots satisfy the Poenaru conjecture. Topology Appl.18, 235–258 (1984)
Schreier, O.: Über die GruppenA a B b=1. Abh. Math. Semin. Univ. Hamb.3, 167–169 (1924)
Schubert, H.: Knoten mit zwei Brücken. Math. Z.65, 133–170 (1956)
Takahashi, M.: Two-bridge knots have PropertyP. Memoirs Am. Math. Soc.29 (1981)
Takeuchi, Y.: The untangling theorem. Topology Appl.34, 129–137 (1990)
Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc.19, 417–431 (1988)
Uchida, Y.: Detecting inequivalent of the unknotting tunnels for a two-bridge knot. Preprint
Zieschang, H.: Generators of the free product with amalgamation of two infinite cyclic groups. Math. Ann.277, 195–221 (1977)
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Dedicated to Professor Yoko Tao for her 60th birthday
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Morimoto, K., Sakuma, M. On unknotting tunnels for knots. Math. Ann. 289, 143–167 (1991). https://doi.org/10.1007/BF01446565
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DOI: https://doi.org/10.1007/BF01446565