Knot manifolds with isomorphic spines

• Autorzy: Alberto Cavicchioli , Friedrich Hegenbarth
• Języki publikacji: EN

Abstrakty

EN

We study the relation between the concept of spine and the representation of orientable bordered 3-manifolds by Heegaard diagrams. As a consequence, we show that composing invertible non-amphicheiral knots yields examples of topologically different knot manifolds with isomorphic spines. These results are related to some questions listed in [9], [11] and recover the main theorem of [10] as a corollary. Finally, an application concerning knot manifolds of composite knots with h prime factors completes the paper.

Słowa kluczowe

EN

3-manifold; spine; group presentation; Heegaard diagram; knot; knot group; knot manifold; peripheral system

Kategorie tematyczne

• 57Q40: Regular neighborhoods
• 57M25: Knots and links in S 3
• 57N10: Topology of general 3 -manifolds
• 57M05: Fundamental group, presentations, free differential calculus

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1994
• Tom: 145
• Numer: 1
• Strony: 79-89

Daty

wydano

1994

otrzymano

1993-06-17

Twórcy

autor

Alberto Cavicchioli

• Dipartimento di Matematica, Università di Modena, via Campi 213/b, 41100 Modena, Italy

autor

Friedrich Hegenbarth

• Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italy

Bibliografia

• [1] G. Burde and H. Zieschang, Knots, Walter de Gruyter, Berlin, 1985.
• [2] A. Cavicchioli, Imbeddings of polyhedra in 3-manifolds, Ann. Mat. Pura Appl. 162 (1992), 157-177.
• [3] R. Craggs, Free Heegaard diagrams and extended Nielsen transformations, I, Michigan Math. J. 26 (1979), 161-186; II, Illinois J. Math. 23 (1979), 101-127.
• [4] M. Culler, C. Mc A. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Bull. Amer. Math. Soc. 13 (1985), 43-45; Ann. of Math. 125 (1987), 237-300.
• [5] C. D. Feustel and W. Whitten, Groups and complement of knots, Canad. J. Math. 30 (1978), 1284-1295.
• [6] C. Mc A. Gordon and J. Luecke, Knots are determined by their complements, Bull. Amer. Math. Soc. 20 (1989), 83-87; J. Amer. Math. Soc. 2 (1989), 371-415.
• [7] J. Hempel, 3-manifolds, Princeton Univ. Press, Princeton, N.J., 1976.
• [8] L. H. Kauffman, On knots, Ann. of Math. Stud. 115, Princeton Univ. Press, Princeton, N.J., 1987.
• [9] R. Kirby, Problems in low dimensional manifold theory, in: Proc. Sympos. Pure Math. 32, Amer. Math. Soc., Providence, R.I., 1978, 273-312.
• [10] W. J. R. Mitchell, J. Przytycki and D. Repovš, On spines of knot spaces, Bull. Polish Acad. Sci. 37 (1989), 563-565.
• [11] D. Repovš, Regular neighbourhoods of homotopically PL embedded compacta in 3-manifolds, Suppl. Rend. Circ. Mat. Palermo 18 (1988), 415-422.
• [12] D. Rolfsen, Knots and Links, Math. Lecture Ser. 7, Publish or Perish, Berkeley, 1976.
• [13] T. B. Rushing, Topological Embeddings, Academic Press, New York, 1973.
• [14] J. Simon, On the problems of determining knots by their complements and knot complements by their groups, Proc. Amer. Math. Soc. 57 (1976), 140-142.
• [15] J. Simon, How many knots have the same group?, ibid. 80 (1980), 162-166.
• [16] J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), 88-111.
• [17] J. Stillwell, Classical Topology and Combinatorial Group Theory, Springer, New York, 1980.
• [18] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56-88.
• [19] W. Whitten, Rigidity among prime-knot complements, Bull. Amer. Math. Soc. 14 (1986), 299-300.
• [20] W. Whitten, Knot complements and groups, Topology 26 (1987), 41-44.
• [21] P. Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208 (1975), 161-169.
• [22] E. C. Zeeman, Seminar on Combinatorial Topology, mimeographed notes, Inst. des Hautes Études Sci., Paris, 1963.