Virtual knot invariants arising from parities

Denis Petrovich Ilyutko, Vassily Olegovich Manturov, and Igor Mikhailovich Nikonov. "Virtual knot invariants arising from parities." Banach Center Publications 100.1 (2014): 99-130. <http://eudml.org/doc/282411>.

@article{DenisPetrovichIlyutko2014,
abstract = { 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. },
author = {Denis Petrovich Ilyutko, Vassily Olegovich Manturov, Igor Mikhailovich Nikonov},
journal = {Banach Center Publications},
keywords = {Gaussian parity; virtual knot invariant; free knot; flat knot; universal parity; homological parity; parity bracket},
language = {eng},
number = {1},
pages = {99-130},
title = {Virtual knot invariants arising from parities},
url = {http://eudml.org/doc/282411},
volume = {100},
year = {2014},
}

TY - JOUR
AU - Denis Petrovich Ilyutko
AU - Vassily Olegovich Manturov
AU - Igor Mikhailovich Nikonov
TI - Virtual knot invariants arising from parities
JO - Banach Center Publications
PY - 2014
VL - 100
IS - 1
SP - 99
EP - 130
AB - 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.
LA - eng
KW - Gaussian parity; virtual knot invariant; free knot; flat knot; universal parity; homological parity; parity bracket
UR - http://eudml.org/doc/282411
ER -