Parity and cobordism of free knots
type: Item
instance of: scholarly article
title: Parity and cobordism of free knots
zbMATH Open document ID: 1259.05047
published in: Sbornik: Mathematics
publication date: 2012-05-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.2827
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6039347
zbMATH Keywords: knot
zbMATH Keywords: free knot
zbMATH Keywords: cobordism
zbMATH Keywords: parity
zbMATH Keywords: surface
author: V. O. Manturov
MaRDI profile type: Publication
arXiv ID: 1001.2827
OpenAlex ID: W1997960508
DOI: 10.1070/SM2012V203N02ABEH004219
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review text: For an oriented long free knot \(K\), the author defines an invariant \(l(K)\) and \(L(K)=|l(K)|\) by using \textit{refined parities} due to himself [Sb. Math. 201, No. 5, 693--733 (2010; Zbl 1210.57010)]. Then \(l(K)\) detects non-invertibility as a long free knot (Corollary 2), and \(L(K)\) is an invariant as an oriented/unoriented free knot (Corollary 3). An example is raised which shows \(L(K)\) is a non-trivial invariant (Figure 7). The main theorem of the present paper is that \(L(K)\) is a cobordism invariant, and \(L(K)\neq 0\) implies \(K\) is not null-cobordant (\(=\) is non-trivial) (Theorem 4 in Section 6).NEWLINENEWLINEA \textit{free knot} and a \textit{free link} are the same notion as a \textit{Gauss word} and a \textit{Gauss phrase} respectively in nanoword theory due to \textit{V.~Turaev} (for example, see [Int. Math. Res. Not. 2006, No. 25, Article ID 84098, 23 p. (2006; Zbl 1118.57009)]), which forget information of the local intersection number at every crossing from a flat virtual knot and a flat virtual link. A free knot is also regarded as a chord diagram and a framed \(4\)-valent graph. Here the meaning of `framed' is that every \(4\)-valent vertex is rigid as an intersection of two local arcs but the sign (the local intersection number) is not determined even if it is oriented and base-pointed. The author uses them effectively. A chord diagram is used when equivalence relations induced from Reidemeister moves are defined (Subsection 1.2), and when notions of the \textit{gaussian parity} and the \textit{refined gaussian parity} of every crossing is introduced (Definition 4 in Subsection 1.3). A framed \(4\)-valent graph is used when a notion of \textit{cobordism} is introduced (Section 4).NEWLINENEWLINEIn Definition 2 and 3 in Subsection 1.3, notions of the gaussian parity and the refined gaussian parity for a free knot are generalized as a \textit{parity} and a \textit{refined parity}, respectively. A parity \(p(v)\) of a crossing \(v\) is v…
reviewed by: Teruhisa Kadokami
description: scientific article; zbMATH DE number 6039347