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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



Compact and long virtual knots

Author: V. O. Manturov
Translated by: G. G. Gould
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 69 (2008).
Journal: Trans. Moscow Math. Soc. 2008, 1-26
MSC (2000): Primary 57M25, 57M27; Secondary 57R65
Published electronically: November 19, 2008
MathSciNet review: 2549444
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Abstract: The theory of virtual knots is a generalization of the theory of classical knots proposed by Kauffman in 1996. In this paper solutions of certain problems of the theory of virtual knots are given. The first part of the paper is devoted to the representation of virtual knots as knots in 3-manifolds of the form $ S_g\times I$ modulo stabilization. Using Haken's theory of normal surfaces, a theorem is proved showing that the problem of recognizing virtual links is algorithmically soluble. A theorem is proved stating that (any) connected sum of non-trivial virtual knots is non-trivial. This result is a corollary of inequalities proved in this paper for the (underlying) genus of virtual knots. The last part of this paper is concerned with long virtual knots. By contrast with the classical case, the classifications of long and compact virtual knots are different. Invariants of long virtual knots are constructed with the help of which one can establish the inequivalence of long knots having equivalent closures and also that some pairs of long virtual knots do not commute.

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Additional Information

V. O. Manturov
Affiliation: Eniseiskaya Ulitsa, 16/21–34, Moscow 129344, Russia

Keywords: Classical knot, long virtual knot, classical crossing, virtual crossing, thickened surface, connected sum, long groupoid, Haken manifold, Reidemeister motion, destabilization
Published electronically: November 19, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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