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1. Positive braids minimize ascending number

   https://doi.org/10.1142/S0218216525500725  

   Davis, Lowell; Meier, Jeffrey (January 2026, Journal of Knot Theory and Its Ramifications)

   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. 

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2. An effective Chabauty–Kim theorem

   https://doi.org/10.1112/S0010437X19007243  

   Balakrishnan, Jennifer S.; Dogra, Netan (June 2019, Compositio Mathematica)

   The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem. 

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3. Knot Floer homology and the unknotting number

   Alishahi, A.; Eftekhary, E. (January 2021, Geometry topology)

   Given a knot K in S3, let u−(K) (respectively, u+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l−(K),l+(K) and l(K), which give lower bounds on u−(K),u+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and satisfies l(K) ≥ ν+(K), while the difference l(K) − ν+(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance. 

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4. Quadratic Chabauty for modular curves: algorithms and examples

   https://doi.org/10.1112/S0010437X23007170  

   Balakrishnan, Jennifer S.; Dogra, Netan; Müller, J. Steffen; Tuitman, Jan; Vonk, Jan (June 2023, Compositio Mathematica)

   We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$$g>1$$whose Jacobians have Mordell–Weil rank$$g$$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$$X_0^+(N)$$of prime level$$N$$, the curve$$X_{S_4}(13)$$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$$X_{\scriptstyle \mathrm { ns}} ^+ (17)$$. 

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5. Optimal angle bounds for Steiner triangulations of polygons

   https://doi.org/10.1137/1.9781611977073.121  

   Christopher J. Bishop (January 2022, Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))

   For any simple polygon P we compute the optimal upper and lower angle bounds for triangulating P with Steiner points, and show that these bounds can be attained (except in one special case). The sharp angle bounds for an N-gon are computable in time O(N), even though the number of triangles needed to attain these bounds has no bound in terms of N alone. In general, the sharp upper and lower bounds cannot both be attained by a single triangulation, although this does happen in some cases. For example, we show that any polygon with minimal interior angle θ has a triangulation with all angles in the interval I = [θ, 90°–min(36°, θ)/2], and for θ ≤ 36° both bounds are best possible. Surprisingly, we prove the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. The proof of this verifies, in a stronger form, a 1984 conjecture of Gerver. 

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