The unknotting number and band-unknotting number of a knot
type: Item
instance of: scholarly article
title: The unknotting number and band-unknotting number of a knot
zbMATH Open document ID: 1255.57002
author: Tetsuya Abe
author: Ryo Hanaki
author: Ryuji Higa
published in: Osaka Journal of Mathematics
publication date: 2012-07-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1340197938
reviewed by: Chuichiro Hayashi
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 6060698
zbMATH Keywords: unknotting number
zbMATH Keywords: unknotting number of a knot diagram
zbMATH Keywords: crossing change
zbMATH Keywords: braid index
zbMATH Keywords: positive braid
zbMATH Keywords: band-move
zbMATH Keywords: H(2)-move
zbMATH Keywords: band-unknotting number
zbMATH Keywords: twist knot
MaRDI profile type: Publication
cites work: A note on knots with \(\mathrm{H}(2)\)-unknotting number one
cites work: A note on unknotting number
cites work: Applications of Donaldson's theorems to classical knot concordance, homology 3-spheres and property P
cites work: Pseudo diagrams of knots, links and spatial graphs
cites work: ON AN INEQUALITY BETWEEN UNKNOTTING NUMBER AND CROSSING NUMBER OF LINKS
cites work: Trivializing number of knots
cites work: CLASSICAL AND VIRTUAL PSEUDODIAGRAM THEORY AND NEW BOUNDS ON UNKNOTTING NUMBERS AND GENUS
cites work: Unknotting operations involving trivial tangles
cites work: Q3052515
cites work: Two-Bridge Knots with Unknotting Number One
cites work: State models and the Jones polynomial
cites work: Unknotting by adding a Twisted Band
cites work: Q4089518
cites work: Jones polynomials and classical conjectures in knot theory
cites work: Four-genus and unknotting number of positive knots and links
cites work: Q3341550
cites work: Positive knots have negative signature
cites work: Lens space surgeries and a conjecture of Goda and Teragaito
cites work: Khovanov homology and the slice genus
cites work: Unknotting number one knots are prime
cites work: Ozsváth Szabó's correction term and lens surgery
cites work: A partial order of knots
cites work: UNKNOTTING NUMBERS OF DIAGRAMS OF A GIVEN NONTRIVIAL KNOT ARE UNBOUNDED
cites work: Q4328310
cites work: A spanning tree expansion of the Jones polynomial
review text: Let \(D\) be a knot diagram with one or more crossings, and let \(c(D)\) the number of crossings of \(D\). It is well-known that by changing over/under information at some crossings of \(D\) we obtain a diagram of the trivial knot. Let \(u(D)\) be the minimal number of such crossing changes. For a knot \(K\), the crossing number \(c(K)\) and the unknotting number \(u(K)\) are the minima of \(c(D)\) and \(u(D)\), where \(D\) is taken over all diagrams of \(K\). As it is well-known, the inequalities \(u(D) \leq (c(D)-1)/2\) and \(u(K) \leq (c(K)-1)/2\) hold. \textit{K. Taniyama} [J. Knot Theory Ramifications 18, No. 8, 1049--1063 (2009; Zbl 1180.57016)] showed that if the equality \(u(D) = (c(D)-1)/2\) holds, then \(D\) is the closure of a positive (negative) \(2\)-braid, and hence, if the equality \(u(K) = (c(K)-1)/2\) holds, then \(K\) is a \((2,r)\)-torus knot for some odd integer \(r \neq \pm1\).NEWLINENEWLINEIn the paper under review, Theorem 2.12 states that \(u(D) = (c(D)-2)/2\) holds for a reduced knot diagram \(D\) if and only if \(D\) is the diagram of the figure-eight knot with four crossings, the closure of a positive (negative) \(3\)-braid, or the connected sum of two closures of positive or negative \(2\)-braids. Hence, if \(u(K)=(c(K)-2)/2\) for a knot \(K\), then \(K\) has a knot diagram as above and is of braid index \(3\). In Figure 8, the suffix numbers \(1,2,3\) of \(a\) should be \(i,j,k\) with \(i<j<k\).NEWLINENEWLINEReviewer's remark: According to one of the authors, Ryo Hanaki, the proof has a gap in line 3 page 531 concerning Figure 8 and he has a proof filling it in his brain.NEWLINENEWLINELet \(D\) be a knot diagram, and \(H\) a square in the projection plane such that \(H\) intersects \(D\) in a single pair of opposite edges. Then the deformation obtaining \((D - H) \cup (\partial H - D)\) from \(D\) is called a band-move. The band-unknotting number \(u_b(K)\) of a knot \(K\) is the minimal number of band-moves to deform a diagram of \(K\) …
description: scientific article; zbMATH DE number 6060698