Knots, butterflies and 3-manifolds..

H. M. Hilden; J. M. Montesinos; D. M. Tejada; M. M. Toro

Displaying similar documents to “Knots, butterflies and 3-manifolds..”

Geometric types of twisted knots

Mohamed Aït-Nouh, Daniel Matignon, Kimihiko Motegi (2006)

Annales mathématiques Blaise Pascal

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Let $K$ be a knot in the $3$ -sphere $S^{3}$ , and $Δ$ a disk in $S^{3}$ meeting $K$ transversely in the interior. For non-triviality we assume that $| Δ \cap K | \geq 2$ over all isotopies of $K$ in $S^{3} - \partial Δ$ . Let $K_{Δ, n}$ ( $\subset S^{3}$ ) be a knot obtained from $K$ by $n$ twistings along the disk $Δ$ . If the original knot is unknotted in $S^{3}$ , we call $K_{Δ, n}$ a . We describe for which pair $(K, Δ)$ and an integer $n$ , the twisted knot $K_{Δ, n}$ is a torus knot, a satellite knot or a hyperbolic knot.

Basic polyhedra in knot theory

Slavik V. Jablan, Ljiljana M. Radović, Radmila Sazdanović (2005)

Kragujevac Journal of Mathematics

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Invariants of piecewise-linear knots

Richard Randell (1998)

Banach Center Publications

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We study numerical and polynomial invariants of piecewise-linear knots, with the goal of better understanding the space of all knots and links. For knots with small numbers of edges we are able to find limits on polynomial or Vassiliev invariants sufficient to determine an exact list of realizable knots. We thus obtain the minimal edge number for all knots with six or fewer crossings. For example, the only knot requiring exactly seven edges is the figure-8 knot.

Intrinsic knotting and linking of complete graphs.

Flapan, Erica (2002)

Algebraic & Geometric Topology

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On a problem in effective knot theory

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 $F (n) : Z \to Z$ such that if $Γ$ is a knot diagram with $n$ 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 $F (n)$ crossings». The problem is shown to be equivalent to a problem posed by D. Welsh in [7] and solved by geometrical techniques (normal surfaces).

On the impossibility of a generalization of the HOMFLY — Polynomial to Labelled Oriented Graphs

Antje Christensen, Stephan Rosebrock (1996)

Annales de la Faculté des sciences de Toulouse : Mathématiques

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Coloured knots and permutations representing 3-manifolds.

Grasselli, Luigi (1996)

Mathematica Pannonica

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Generalized n-colorings of links

Daniel Silver, Susan Williams (1998)

Banach Center Publications

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The notion of an (n,r)-coloring for a link diagram generalizes the idea of an n-coloring introduced by R. H. Fox. For any positive integer n the various (n,r)-colorings of a diagram for an oriented link l correspond in a natural way to the periodic points of the representation shift $Φ_{/ n} (l)$ of the link. The number of (n,r)-colorings of a diagram for a satellite knot is determined by the colorings of its pattern and companion knots together with the winding number.