Noncommutative knot theory.
Cochran, Tim D. (2004)
Algebraic & Geometric Topology
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Cochran, Tim D. (2004)
Algebraic & Geometric Topology
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Józef Przytycki (1995)
Banach Center Publications
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We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical...
Lieberum, Jens (2002)
International Journal of Mathematics and Mathematical Sciences
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Stephen Budden, Roger Fenn (2004)
Fundamenta Mathematicae
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Let A, B be invertible, non-commuting elements of a ring R. Suppose that A-1 is also invertible and that the equation [B,(A-1)(A,B)] = 0 called the fundamental equation is satisfied. Then this defines a representation of the algebra ℱ = A, B | [B,(A-1)(A,B)] = 0. An invariant R-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when R is the quaternions. ...
Friedl, Stefan, Teichner, Peter (2005)
Geometry & Topology
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Nafaa Chbili (2003)
Annales de la Faculté des sciences de Toulouse : Mathématiques
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Livingston, Charles (2002)
Algebraic & Geometric Topology
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Paweł Traczyk (1995)
Banach Center Publications
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Ng, Lenhard L. (2001)
Algebraic & Geometric Topology
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Alexander Stoimenow (2003)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Using the recent Gauß diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no...