Three-page approach to knot theory. Encoding and local moves
A tame link in \(\mathbb R^3\) can be embedded in finitely many pages in an open-book decomposition of \(\mathbb R^3\) so that it meets each page in a single simple arc. Such an embedding of a knot or link is called an arc-presentation. This kind of presentation was discussed very early and, recently, \textit{J. S. Birman} and \textit{W. W. Menasco} reinvented this presentation [Topology 33, No. 3, 525-556 (1994; Zbl 0833.57004)]. The properties of this new presentation of knots and links have been discussed. For example, see [\textit{H. R. Morton} and \textit{E. Beltrami}, Math. Proc. Camb. Philos. Soc. 123, No. 1, 41-48 (1998; Zbl 0888.57007); \textit{I. J. Nutt}, ibid. 126, No. 1, 77-98 (1999; Zbl 0918.57003)]. In the paper under review, the author focuses on three-page link diagrams and shows that this restriction gives a new combinatorial way to study knots and links. A three-page link diagram is a closed polygonal line \(L\) embedded in an open book decomposition of Euclidean \(3\)-space \(R^3\) with three pages \(P_1, P_2,\) and \(P_3\) satisfying the two conditions: (i) the intersection \(L\cap \ell\) is a finite point set, \(L\cap \ell = \{A_1, \dots, A_m\},\) and two edges of \(L\) attached to \(A_k\) lie in different pages for any \(k\leq m\); (ii) the restriction of the orthogonal projection \(R^3 \to \ell \cong R\) to any connected component of \(L\cap P_i\) is monotone function for any \(i \in \{1, 2, 3\}.\) The author shows that any knot and link can be represented by a three-page link diagram and that two three-page diagrams define isotopic links if and only if they can be obtained from each other by a finite sequence of local moves of three types analogue to the Reidemeister moves for planar link diagrams. More interesting is that the author gives a way to encode links by words in a finite alphabet and shows that two words define isotopic nonoriented links in \(\mathbb R^3\) if and only if one can be obtained from the other by a finite sequence of local changes of subwords. For further results see [the author, Three-page approach to knot theory. Universal semigroup, Funct. Anal. Appl. 34, No. 1, 24-32 (2000); translation from Funkts. Anal. Prilozh. 34, No. 1, 29-40 (2000)].
scientific article; zbMATH DE number 1452674