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A well-known algorithm for unknotting knots involves traversing a knot diagram and changing each crossing that is first encountered from below. The minimal number of crossings changed in this way across all diagrams for a knot is called the ascending number of the knot. The ascending number is bounded below by the unknotting number. We show that for knots obtained as the closure of a positive braid, the ascending number equals the unknotting number. We also present data indicating that a similar result may hold for positive knots. We use this data to examine which low-crossing knots have the property that their ascending number is realized in a minimal crossing diagram, showing that there are at most 5 hyperbolic, alternating knots with at most 12 crossings with this property. 

Abstract We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise nonisotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive answer to a stronger version of an old question of Hitt and Sumners. The examples arise from our main result: a classification of fibered, homotopy‐ribbon disks for each generalized square knot up to isotopy. Precisely, we show that each generalized square knot bounds infinitely many pairwise nonisotopic fibered, homotopy‐ribbon disks, all of whose exteriors are diffeomorphic. When , we prove further that infinitely many of these disks are also ribbon; whether the disks are always ribbon is an open problem. 

The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and Zupan. We show that the set of Alexander polynomials of knots with ribbon number at most four contains 56 polynomials, and we use this set to compute the ribbon numbers for many 12-crossing knots. We also study higher-genus ribbon numbers of knots, presenting some examples that exhibit interesting behavior and establishing that the success of the Alexander polynomial at controlling genus-0 ribbon numbers does not extend to higher genera. 

Abstract A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $$\mathbb{R}^n$$ for some n , sending E to a starlike set. A subset $$E\subset X$$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $$\{E_i\}_{i=0}^{N+1}$$ such that $$E_{i}/E_{i+1}\subset X/E_{i+1}$$ is starlike-equivalent for each i and $$E_{N+1}$$ is a point. A decomposition $$\mathcal{D}$$ of a metric space X is said to be recursively starlike-equivalent, if there exists $$N\geq 0$$ such that each element of $$\mathcal{D}$$ is recursively starlike-equivalent of filtration length N . We prove that any null, recursively starlike-equivalent decomposition $$\mathcal{D}$$ of a compact metric space X shrinks, that is, the quotient map $$X\to X/\mathcal{D}$$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture. 

Abstract We study embedded spheres in 4–manifolds (2–knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilisation and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2–knots directly from doubly pointed trisection diagrams. The operations described are classical surgery, Gluck surgery, blowdown, and (±4)–rational blowdown, and we illustrate our techniques and results with many examples.