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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2011, Volume 17, Number 4, Pages 160–175
(Mi timm761)
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This article is cited in 3 scientific papers (total in 3 papers)
Uniqueness of knot roots in $F\times I$ and prime decompositions of virtual knots
F. G. Korablevab a Chelyabinsk State University
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
We introduce four types of reduction on the set of knots in thickened surfaces, i.e., in three-dimensional manifolds of the form $F\times I$, where $F$ is a closed orientable surface and $I=[0,1]$. It is proved that the process of applying these reductions to an arbitrary knot in a thickened surface is always finite. The resulting set of knots in thickened surfaces depends on the initial knot only up to the removal of trivial knots in thickened spheres. Reductions of knots in thickened surfaces induce the operation of connected summation of virtual knots. It is proved that every virtual knot can be decomposed into a connected sum of several prime or trivial virtual knots and the prime summands of the decomposition are defined uniquely.
Keywords:
virtual knot, root theory, connected sum, thickened surface.
Received: 09.04.2011
Citation:
F. G. Korablev, “Uniqueness of knot roots in $F\times I$ and prime decompositions of virtual knots”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 4, 2011, 160–175
Linking options:
https://www.mathnet.ru/eng/timm761 https://www.mathnet.ru/eng/timm/v17/i4/p160
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Abstract page: | 282 | Full-text PDF : | 67 | References: | 36 | First page: | 1 |
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