Representation of -knots.
Mulazzani, Michele (2006)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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Mulazzani, Michele (2006)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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Perko, Kenneth A. jr. (1979)
Portugaliae mathematica
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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:...
Livingston, Charles (2002)
Algebraic & Geometric Topology
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A. Stoimenow (2008)
Fundamenta Mathematicae
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We give a description of all knot diagrams of canonical genus 2 and 3, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic volume for canonical (weak) genus 2 knots. We also study the values of the link polynomials at roots of unity, extending denseness results of Jones. Using...
Clark, Bradd Evans (1983)
International Journal of Mathematics and Mathematical Sciences
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Livingston, Charles (2004)
Geometry & Topology
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Steven A. Bleiler, Yoav Moriah (1988)
Mathematische Annalen
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Livingston, Charles (2004)
Algebraic & Geometric Topology
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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.
C. Kearton, E. Bayer-Fluckiger (1989)
Mathematische Zeitschrift
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Daniel S. Silver, Susan G. Williams (2009)
Banach Center Publications
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A conjecture of [swTAMS] states that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.
Mohamed Ait Nouh, Akira Yasuhara (2001)
Revista Matemática Complutense
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We give a necessary condition for a torus knot to be untied by a single twisting. By using this result, we give infinitely many torus knots that cannot be untied by a single twisting.
Livingston, Charles (2003)
Geometry & Topology
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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.