On a problem of effective knot theory
A knot is an embedded circle in the Euclidean space \(\mathbb R^3\). A knot is trivial if it bounds an embedded disk. A knot diagram is an image of a knot by the projection \(\mathbb R^3 \rightarrow \mathbb R^2\) with double points (called crossings) assigned ``upper'' and ``under''. A diagram is trivial if it has no crossings. Two diagrams of a knot are connected by a finite sequence of local moves called Reidemeister moves [\textit{J. W. Alexander} and \textit{G. B. Briggs}, Ann. Math. (2) 28, 562--586 (1927; JFM 53.0549.02); \textit{H. Reidemeister}, Abh. Math. Sem. Univ. Hamburg 5, 24--32 (1926; JFM 52.0579.01)]. The unknotting problem, i.e., to decide whether a given knot diagram represents the trivial knot or not is difficult in practice [see \textit{L. Goeritz}, Abh. Math. Semin. Hamb. Univ. 10, 201--210 (1934; Zbl 0009.23001)]. For diagrams of non-trivial knots and links, see pages 11, 21 and 22 in [\textit{M. B. Thistlethwaite}, Lond. Math. Soc. Lect. Note Ser. 93, 1--76 (1985; Zbl 0571.57004)] and [Topology and Computer Science, 225--249 (Kinokuniya, Tokyo) (1987)] by \textit{S. Nagami}. \textit{W. Haken} gave a finite algorithm for the unknotting problem [Acta Math. 105, 245--375 (1961; Zbl 0100.19402)]. His idea is based on a theory of fundamental normal surfaces in \(3\)-manifolds with handle decompositions, which was formulated in triangulated \(3\)-manifolds by \textit{W. Jaco} and \textit{U. Oertel} [Topology 23, No. 2, 195--209 (1984; Zbl 0545.57003)]. See also a survey [Chaos Solitons Fractals 9, No. 4-5, 569--581 (1998; Zbl 0935.57014)] by \textit{J. Hass}, and another algorithm [Geom. Topol. 2, 175--220 (1998; Zbl 0955.57005)] by \textit{J. S. Birman} and \textit{M. D. Hirsch}. The paper under review gives an elementary function \(f(n)\) such that any knot diagram with \(n\) crossings is converted to the trivial diagram by a sequence of no more than \(f(n)\) Reidemeister moves if it represents the trivial knot, with a sketch proof based on the machinery of Jaco and Oertel. Similar results are obtained by \textit{J. Hass} and \textit{J. C. Lagarias} [The number of Reidemeister moves needed for unknotting, J. Am. Math. Soc. 14, No. 2, 399--428 (2001; Zbl 0964.57005)] and by \textit{J. Hass, J. C. Lagarias} and \textit{N. Pippenger} [The computational complexity of knot and link problems. J. ACM 46, No. 2, 185--211 (1999; Zbl 1065.68667)].
scientific article; zbMATH DE number 1322985