Hass, Joel The geometry of the slice-ribbon problem. (English) Zbl 0535.57004 Math. Proc. Camb. Philos. Soc. 94, 101-108 (1983). A ribbon knot is a knot which can be isotoped to an immersed disk consisting of a number of disks with bands and the only intersection being bands cutting transversely through the interior of a disk. A slice knot is a knot which bounds a disk properly embedded in a 4-ball. The question of whether all slice knots are ribbon knots is still open. This paper gives some geometric descriptions of ribbon knots. A ribbon disk is defined as an embedded \(D^ 2\) in \(B^ 4\) with the property that the function \(x\mapsto r(x)\) defined by taking the distance from the center of \(B^ 4\) restricts to a Morse function on the disk with no critical points of index 2. In a lemma, it is shown that a knot is ribbon if and only if it bounds a ribbon disk. Using a notion of total curvature based on the Lipschitz-Killing curvature, the author shows the key result of the paper: An embedded disk F in \(B^ 4\) of total curvature less than \(4\pi^ 2\) can be isotoped to a ribbon disk. The converse is easier. From this proposition, the following are shown to be equivalent for a knot k: 1) k is ribbon. 2) k bounds an embedded minimal disk in \(B^ 4\). 3) k bounds an embedded disk of curvature less than \(4\pi^ 2\). 4) k bounds an embedded ruled disk in \(B^ 4\). Reviewer: G.E.Lang Cited in 9 Documents MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010) 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C40 Global submanifolds 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature Keywords:ribbon knot; slice knot; ribbon disk; Morse function; total curvature; minimal disk; ruled disk × Cite Format Result Cite Review PDF Full Text: DOI References: [1] DOI: 10.1007/BF01214308 · Zbl 0479.49026 · doi:10.1007/BF01214308 [2] DOI: 10.2307/1969467 · Zbl 0037.38904 · doi:10.2307/1969467 [3] DOI: 10.2307/2372684 · Zbl 0078.13901 · doi:10.2307/2372684 [4] Kirby, Proc. A.M.S. Summer Institute in Topology (1976) [5] Fox, Proc. Univ. Georgia Inst pp 120– (1962) [6] DOI: 10.1016/0040-9383(76)90034-3 · Zbl 0341.53037 · doi:10.1016/0040-9383(76)90034-3 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.