Unknotting via null-homologous twists and multitwists
type: Item
instance of: scholarly article
title: Unknotting via null-homologous twists and multitwists
author: Samantha Allen
author: Kenan Ince
author: Seung Won Kim
author: Benjamin Matthias Ruppik
author: Hannah Turner
published in: Pacific Journal of Mathematics
publication date: 2024-07-29T00:00:00Z
review text: 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) \le…
reviewed by: Keiko Kawamuro
Mathematics Subject Classification ID: 57K10
Mathematics Subject Classification ID: 57K40
zbMATH DE Number: 7887953
zbMATH Keywords: 4-manifolds
zbMATH Keywords: surgery diagram
zbMATH Keywords: unknotting operation
zbMATH Keywords: untwisting number
MaRDI profile type: Publication
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DOI: 10.2140/PJM.2024.330.25
description: scientific article; zbMATH DE number 7887953