The unknotting number and band-unknotting number of a knot
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 \(K\) into that of the unknot by Reidemeister moves and band-moves. It is shown that, if \(u_b(K)=1\) for a twist knot \(K\), then \(K=3_1, 5_2, 6_1\) or \(7_2\) up to mirror images. \textit{K. Taniyama} and \textit{A. Yasuhara} [Kobe J. Math. 11, No. 1, 117--127 (1994; Zbl 0846.57007)] showed that \(u_b(K) \leq c(K)/2\) for every knot \(K\). The paper under review shows that, if \(u_b(K)= c(K)/2\), then \(K\) is the trivial knot or the figure-eight knot. To show this, the inequality \(u_b(K) \leq u(K) + 1\) is used.
scientific article; zbMATH DE number 6060698
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