Unknotting via null-homologous twists and multitwists
The crossing change is a basic unknotting operation, and the minimal number of crossing changes needed to obtain an unknot is called the \textit{unknotting number} \(\mathrm{u}(K)\). A \textit{null-homologous twist}, defined by \textit{Y. Mathieu} and \textit{M. Domergue} [Math. Ann. 280, No. 3, 501--508 (1988; Zbl 0618.57004)], is a generalization of the unknotting operation and is the result of a \(\pm 1\)-surgery on an unknot \(U \subset S^3 \setminus K\) bounding a disk \(D\) such that \(|D \cap K| = 2k\) and \(\mathrm{lk}(K, U)=0\). The \textit{untwisting number} \(\mathrm{tu}(K)\) the minimal number of null-homologous twists applied to \(K\) to obtain an unknot. The \textit{surgery description number} \(\mathrm{sd}(K)\), due to \textit{Y. Nakanishi} [J. Knot Theory Ramifications 14, No. 1, 3--8 (2005; Zbl 1067.57005)], is the minimal number of null-homologous \(m\)-twists (here, \(m\) can be any integer and may change from move to move).\N\NFor each invariant, we can consider the minimal number of the operations needed to obtain an Alexander polynomial-one knot and define the \textit{algebraic} unknotting number \(\mathrm{u}_a(K)\) (first introduced by \textit{H. Murakami} [Quest. Answers Gen. Topology 8, No. 1, 283--292 (1990; Zbl 0704.57004)]), \textit{algebraic} untwisting number \(\mathrm{tu}_a(K)\), and \textit{algebraic} surgery description number \(\mathrm{sd}_a(K)\).\N\NThe second author had already shown that \(\mathrm{u}(K)\) and \(\mathrm{tu}(K)\) can be arbitrarily different [\textit{K. Ince}, Pac. J. Math. 283, No. 1, 139--156 (2016; Zbl 1345.57012)]. In this paper, the authors find the first infinite examples of knots such that \(\mathrm{sd} \neq \mathrm{tu}\). Their examples \(\{K_n\}\) are pretzel knots and satisfying \(\mathrm{sd}(K_n)=1 \neq 2 = \mathrm{tu}(K_n)\). They then investigate how far apart \(\mathrm{sd}\) and \(\mathrm{tu}\) can be. Their result shows: for every knot \(K\subset S^3\) we have \(\mathrm{sd}(K) \leq \mathrm{tu}(K) \leq 2\mathrm{sd}(K)+1\), which indicates that \(\mathrm{sd}\) and \(\mathrm{tu}\) have a close relationship. As for the algebraic versions, they obtain \(\mathrm{sd}_a(K) \leq \mathrm{tu}_a(K) \leq 2 \mathrm{sd}_a(K)\) using work of \textit{D. McCoy} [Glasg. Math. J. 63, No. 1, 59--65 (2021; Zbl 1455.57013) and MATRIX Book Ser. 4, 147--165 (2021; Zbl 1498.57006)].
scientific article; zbMATH DE number 7887953
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