A glimpse at knot theory
Paweł Traczyk (1995)
Banach Center Publications
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Paweł Traczyk (1995)
Banach Center Publications
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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...
Ng, Lenhard L. (2001)
Algebraic & Geometric Topology
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Józef Przytycki (1998)
Banach Center Publications
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Roger Fenn, Louis H. Kauffman, Vassily O. Manturov (2005)
Fundamenta Mathematicae
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The present paper gives a quick survey of virtual and classical knot theory and presents a list of unsolved problems about virtual knots and links. These are all problems in low-dimensional topology with a special emphasis on virtual knots. In particular, we touch new approaches to knot invariants such as biquandles and Khovanov homology theory. Connections to other geometrical and combinatorial aspects are also discussed.
Garoufalidis, Stavros, Levine, Jerome (2001)
Algebraic & Geometric Topology
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Szabó, Zoltán, Ozváth, Peter (2003)
Geometry & Topology
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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.
Ng, Lenhard (2005)
Algebraic & Geometric Topology
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Yoshiyuki Ohyama (1994)
Revista Matemática de la Universidad Complutense de Madrid
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We define a twisting move, an (n,k)-move, on a link diagram and consider the question as to whether or not any two links are equivalent by this move. Moreover we show that any knot can be trivialized by at most twice twisting operations.
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.