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An upper bound on Reidemeister moves. (English) Zbl 1301.57004
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, J. Hass and 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.
The 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 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.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
68W40 Analysis of algorithms
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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