An upper bound on Reidemeister moves (Q2879440)

The paper under review gives a remarkable upper bound for the number of Reidemeister moves changing one diagram to another one of the same link. As a preceding work, \textit{J. Hass} and \textit{J. C. Lagarias} [J. Am. Math. Soc. 14, No. 2, 399--428 (2001; Zbl 0964.57005)] showed that any unknotted knot diagram with \(n\) crossings can be transformed to the trivial knot diagram using at most \(2^{cn}\) Reidemeister moves, where \(c=10^{11}\). Thus the present paper answers the general case. Here is the precise statement. Let \(D_1\) and \(D_2\) be connected diagrams of the same knot or link, and let \(n\) be the sum of their crossing numbers. Then \(D_2\) can be obtained from \(D_1\) by at most \(\mathrm{exp}^{(c^n)}(n)\) Reidemeister moves, where \(c=10^{1,000,000}\). The function \(\mathrm{exp}(n)\) is the exponential function \(2^n\), and \(\mathrm{exp}^{(r)}(n)\) is its \(r\)-fold iteration. This also gives a simple solution to the equivalence problem of links.NEWLINENEWLINEThe argument uses triangulations and Pachner moves. It is known that any two triangulations of a PL manifold are related by a sequence of Pachner moves. The key is a result of \textit{A. Mijatović} [Math. Res. Lett. 12, No. 5--6, 843--856 (2005; Zbl 1083.57028)] which gives an upper bound for the number of Pachner moves required to pass between two triangulations of a link exterior. But Mijatović's result is not sufficient here, so a strengthened version is prepared.