Parity and cobordism of free knots
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 valued in \(\mathbb{Z}/2\mathbb{Z}=\{0, 1\}\): \(v\) is \textit{even} if \(p(v)=0\), and \(v\) is \textit{odd} if \(p(v)=1\). A refined parity divides the odd crossings into two types: \textit{odd of the first type} and \textit{odd of the second type}. The generalized notions are defined relatively and axiomatically. In Section 3, it is pointed out that a parity is equivalent to an element with a certain restriction of the first cohomology class over \(\mathbb{Z}/2\mathbb{Z}\) for the free knot as a \(4\)-valent graph. In Section 2, an invariant of a free knot is defined, which is valued in a group \(G\) generated by \(a, b\) and \(b'\) with certain relations. By corresponding the crossings to \(a, b\) and \(b'\) (an even crossing to \(a\), an odd crossing of the first type to \(b\), and an odd crossing of the second type to \(b'\)), a free knot \(K\) determines the conjugacy class of an element in \(G\) of the form \((bb')^{4m}\)\ \((m\in \mathbb{Z})\). Then \(l(K)=4m\) and \(L(K)=4|m|\). In Section 4, a notion of \textit{cobordism} is introduced. Two free knots \(K\) and \(K'\) are \textit{cobordant} if there exists a singular annulus \(F\) surrounded by \(K\cup (-K')\), where the singular set of \(F\) consists of cusps, double lines and triple points.NEWLINENEWLINEAs an alternative proof of non-triviality of free knots (Gauss words), I strongly recommend \textit{A.~Gibson}'s paper [Math. Ann. 349, No. 4, 871--887 (2011; Zbl 1242.57013)]. The present paper is full of new original ideas. Since free knot theory should be the core part of nanoword theory, I feel a sequence of the author's works is very important.
scientific article; zbMATH DE number 6039347