The Homotopoy Type of Four-Dimensional Knot Complements.
Steven P. Plotnick (1983)
Mathematische Zeitschrift
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Steven P. Plotnick (1983)
Mathematische Zeitschrift
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Alessia Cattabriga, Michele Mulazzani (2005)
Fundamenta Mathematicae
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We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG₂(T). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω:PMCG₂(T) → MCG(T) ≅ SL(2,ℤ), which is a free group of rank two, to the class of all (1,1)-knots in a fixed lens space. The second representation is parametric:...
Maciej Borodzik, Krzysztof Oleszkiewicz (2010)
Bulletin of the Polish Academy of Sciences. Mathematics
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We study properties of the signature function of the torus knot . First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.
Dugopolski, Mark J. (1985)
International Journal of Mathematics and Mathematical Sciences
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S. Jablan, R. Sazdanovic (2003)
Visual Mathematics
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Yasutaka Nakanishi (1996)
Revista Matemática de la Universidad Complutense de Madrid
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This note is a continuation of a former paper, where we have discussed the unknotting number of knots with respect to knot diagrams. We will show that for every minimum-crossing knot-diagram among all unknotting-number-one two-bridge knot there exist crossings whose exchange yields the trivial knot, if the third Tait conjecture is true.
Hendricks, Jacob (2004)
Algebraic & Geometric Topology
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Schmitt, Peter (1997)
Beiträge zur Algebra und Geometrie
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Vaughan Jones, Józef Przytycki (1998)
Banach Center Publications
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We show that Lissajous knots are equivalent to billiard knots in a cube. We consider also knots in general 3-dimensional billiard tables. We analyse symmetry of knots in billiard tables and show in particular that the Alexander polynomial of a Lissajous knot is a square modulo 2.
P. V. Koseleff, D. Pecker (2014)
Banach Center Publications
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We show that every knot can be realized as a billiard trajectory in a convex prism. This proves a conjecture of Jones and Przytycki.
Seiichi Kamada (2014)
Banach Center Publications
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A symmetric quandle is a quandle with a good involution. For a knot in ℝ³, a knotted surface in ℝ⁴ or an n-manifold knot in , the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a diagram.
Corinne Cerf (2002)
Visual Mathematics
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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...
Mulazzani, Michele (2006)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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Roger Fenn, Denis P. Ilyutko, Louis H. Kauffman, Vassily O. Manturov (2014)
Banach Center Publications
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This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.
Akira Yasuhara (1992)
Revista Matemática de la Universidad Complutense de Madrid
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We investigate the knots in the boundary of the punctured complex projective plane. Our result gives an affirmative answer to a question raised by Suzuki. As an application, we answer to a question by Mathieu.
Ying-Qing Wu (1993)
Mathematische Annalen
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