Virtual knot invariants arising from parities

• Autorzy: Denis Petrovich Ilyutko , Vassily Olegovich Manturov , Igor Mikhailovich Nikonov
• 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.

Kategorie tematyczne

• 57M27: Invariants of knots and 3-manifolds
• 57M25: Knots and links in S 3

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2014
• Tom: 100
• Numer: 1
• Strony: 99-130

Daty

wydano

2014

Twórcy

autor

Denis Petrovich Ilyutko

• Department of Mechanics and Mathematics, Moscow State University, Russia
• Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Russia

autor

Vassily Olegovich Manturov

• Faculty of Science, Peoples' Friendship University of Russia, Russia
• Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Russia

autor

Igor Mikhailovich Nikonov

• 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

Identyfikatory

DOI

10.4064/bc100-0-6