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2014 | 100 | 1 | 99-130
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Virtual knot invariants arising from parities

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Języki publikacji
EN
Abstrakty
EN
In [12, 15] it was shown that in some knot theories the crucial role is played by parity, i.e. a function on crossings valued in {0,1} and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than ℤ₂ and investigate them for different knot theories. For some knot theories we show that there is the universal parity, i.e. such a parity that any other parity factors through it. We realise that in the case of flat knots all parities originate from homology groups of underlying surfaces and, at the same time, allow one to "localise" the global homological information about the ambient space at crossings.
We prove that there is only one non-trivial parity for free knots, the Gaussian parity. At the end of the paper we analyse the behaviour of some invariants constructed for some modifications of parities.
Słowa kluczowe
Rocznik
Tom
100
Numer
1
Strony
99-130
Opis fizyczny
Daty
wydano
2014
Twórcy
  • Department of Mechanics and Mathematics, Moscow State University, Russia
  • Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Russia
  • Faculty of Science, Peoples' Friendship University of Russia, Russia
  • Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Russia
  • Department of Mechanics and Mathematics, Moscow State University, Russia
  • Faculty of Management, National Research University Higher School of Economics, Russia
  • Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Russia
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-bc100-0-6
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