Knot and braid invariants from contact homology. I.
Ng, Lenhard (2005)
Geometry & Topology
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Ng, Lenhard (2005)
Geometry & Topology
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Mohnke, Klaus
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Kitano, Teruaki, Suzuki, Masaaki (2005)
Experimental Mathematics
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Kirk, P., Livingston, C. (2001)
Geometry & Topology
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Ng, Lenhard L. (2001)
Algebraic & Geometric Topology
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Jablan, Slavik V. (1999)
Novi Sad Journal of Mathematics
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Louis H. Kauffman, Vassily O. Manturov (2005)
Fundamenta Mathematicae
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We describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle [KR], [FJK], the virtual quandle [Ma2], the ideas of quaternion biquandles by Roger Fenn and Andrew Bartholomew [BF], the concepts and properties of long virtual knots [Ma10], and other ideas in the interface between classical and virtual knot theory. In the present paper we present a new algebraic construction of virtual...
Louis H. Kauffman, Sofia Lambropoulou (2004)
Fundamenta Mathematicae
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This paper gives a new method for converting virtual knots and links to virtual braids. Indeed, the braiding method given here is quite general and applies to all the categories in which braiding can be accomplished. This includes the braiding of classical, virtual, flat, welded, unrestricted, and singular knots and links. We also give reduced presentations for the virtual braid group and for the flat virtual braid group (as well as for other categories). These reduced presentations...
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...
Nelson, Sam (2005)
Algebraic & Geometric Topology
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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. ...
Isabel Darcy, De Sumners (1998)
Banach Center Publications
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The following is an expository article meant to give a simplified introduction to applications of topology to DNA.