On two-bridge ribbon knots (Q6590031)

For the notions of ribbon knots or slice knots, please see e.g. [\textit{L. H. Kauffman}, On knots. Princeton University Press, Princeton, NJ (1987; Zbl 0627.57002), Chapter V]. The slice-ribbon conjecture is that all slice knots are ribbon. This conjecture has been one of the most famous conjectures in knot theory for many years. See [\textit{A. Kawauchi}, ``Ribbonness on classical link'', Preprint, \url{arXiv:2307.16483}]. \textit{C. Lamm} [Osaka J. Math. 37, No. 3, 537--550 (2000; Zbl 0976.57009)] gave another conjecture, namely that all ribbon knots are represented as symmetric unions.\N\NThe authors show that two-bridge ribbon knots can be represented as a symmetric union. This paper consists of two theorems, Theorem 1.1 and Theorem 1.2. \N\N\textbf{Theorem 1.1.} A two-bridge knot \(K(m^2, mk \pm 1)\) with \(m > k > 0\) and \((m, k) = 1\) is a ribbon knot which admits a symmetric union presentation with a partial knot which is the two-bridge knot \(K(m, k)\).\N\N\textbf{Theorem 1.2.} A two-bridge knot \(K(m^2, d(m \pm 1))\), where \(d > 1\) divides \(2m \mp\N1\) (respectively, \(d > 1\) is odd and divides \(m \pm 1)\), is a ribbon knot which admits a symmetric union presentation with a partial knot which is the two-bridge knot \(K(m, d)\).\N\NFig.~4 and Fig.~5 at the paper's beginning make it easier to understand Theorem 1.1. And Figures 6 through 9 at the beginning make it easier to understand Theorem 1.2. The proof of Theorem 1.2 is technical and the transformation of the figures is very interesting.