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Virtual knot invariants arising from parities

Tom 100 / 2014

Denis Petrovich Ilyutko, Vassily Olegovich Manturov, Igor Mikhailovich Nikonov Banach Center Publications 100 (2014), 99-130 MSC: 57M25, 57M27. DOI: 10.4064/bc100-0-6

Streszczenie

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 $\mathbb{Z}_2$ 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.

Autorzy

  • Denis Petrovich IlyutkoDepartment of Mechanics and Mathematics
    Moscow State University
    Russia
    and
    Delone Laboratory of Discrete and Computational Geometry
    Yaroslavl State University
    Russia
    e-mail
  • Vassily Olegovich ManturovFaculty of Science
    Peoples' Friendship University of Russia
    Russia
    and
    Delone Laboratory of Discrete and Computational Geometry
    Yaroslavl State University
    Russia
    e-mail
  • Igor Mikhailovich NikonovDepartment of Mechanics and Mathematics
    Moscow State University
    Russia
    and
    Faculty of Management
    National Research University Higher School of Economics
    Russia
    and
    Delone Laboratory of Discrete and Computational Geometry
    Yaroslavl State University
    Russia
    e-mail

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