Classification problems in low-dimensional topology
Klaus Johannson (1986)
Banach Center Publications
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Klaus Johannson (1986)
Banach Center Publications
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Clark, Bradd Evans (1978)
International Journal of Mathematics and Mathematical Sciences
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P.B. Shalen, M. Culler (1984)
Inventiones mathematicae
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A. Hatcher, W. Thurston (1985)
Inventiones mathematicae
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Matsumoto, Saburo, Rannard, Richard (2000)
Experimental Mathematics
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L.M. Lopez (1993)
Mathematische Zeitschrift
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Stefano Galatolo (1998)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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The following problem is investigated: «Find an elementary function such that if is a knot diagram with crossings and the corresponding knot is trivial, then there is a sequence of Reidemeister moves that proves triviality such that at each step we have less than crossings». The problem is shown to be equivalent to a problem posed by D. Welsh in [7] and solved by geometrical techniques (normal surfaces).
Akbulut, Selman, Ozbagci, Burak (2001)
Geometry & Topology
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Osamu Kakimizu (1992)
Mathematische Zeitschrift
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Seiichi Kamada (2001)
Fundamenta Mathematicae
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A Wirtinger presentation of a knot group is obtained from a diagram of the knot. T. Yajima showed that for a 2-knot or a closed oriented surface embedded in the Euclidean 4-space, a Wirtinger presentation of the knot group is obtained from a diagram in an analogous way. J. S. Carter and M. Saito generalized the method to non-orientable surfaces in 4-space by cutting non-orientable sheets of their diagrams by some arcs. We give a modification to their method so that one does not need...
Lee Rudolph (1989)
Revista Matemática de la Universidad Complutense de Madrid
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This is a survey (including new results) of relations ?some emergent, others established? among three notions which the 1980s saw introduced into knot theory: quasipositivity of a link, the enhanced Milnor number of a fibered link, and the new link polynomials. The Seifert form fails to determine these invariants; perhaps there exists an ?enhanced Seifert form? which does.
Dye, H.A., Kauffman, Louis H. (2005)
Algebraic & Geometric Topology
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Hitoshi Murakami (2004)
Fundamenta Mathematicae
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I describe how the colored Jones polynomials of the figure-eight knot determine the volumes of the three-manifolds obtained by Dehn surgeries along it, according to my joint work with Y. Yokota.