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Award ID contains: 2104022

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  1. ; (, Journal of the London Mathematical Society)
    Abstract Let $$T$$ be a satellite knot, link, or spatial graph in a 3-manifold $$M$$ that is either $S^3$ or a lens space. Let $$\b_0$$ and $$\b_1$$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $$T$$ has a companion knot $$K$$ (necessarily not the unknot) and wrapping number $$\omega$$ with respect to $$K$$. When $$K$$ is not a torus knot, we show that $$\b_1(T)\geq \omega \b_1(K)$$. There are previously known counter-examples if $$K$$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $$\b_0(T) \geq \omega \b_0(K)$$. We also prove versions of the theorem applicable to when $$T$$ is a "lensed satellite" and when there is a torus separating components of $$T$$. 
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    Free, publicly-accessible full text available August 1, 2026
  2. ; ; ; ; (, Transactions of the American Mathematical Society, Series B)
    We show that if a composite θ-curve has (proper rational) unknotting number one, then it is the order 2 sum of a (proper rational) unknotting number one knot and a trivial θ-curve. We also prove similar results for 2-strand tangles and knotoids. 
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    Free, publicly-accessible full text available March 4, 2026
  3. (, Boletín de la Sociedad Matemática Mexicana)
    We use thin position of Heegaard splittings to give a new proof of Haken's Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann's ``Strong Haken Theorem'': a Heegaard surface for a 3-manifold may be isotoped to intersect a given collection of essential spheres and discs in a single loop each. We also give a reformulation of Casson and Gordon's theorem on weakly reducible Heegaard splittings, showing that they exhibit additional structure with respect to certain incompressible surfaces. This article could also serve as an introduction to the theory of generalized Heegaard surfaces and it includes a careful study of their behaviour under amalgamation. 
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    Free, publicly-accessible full text available March 1, 2026
  4. ; ; ; (, Algebraic & Geometric Topology)
    A bridge trisection of a smooth surface in S4 is a decomposition analogous to a bridge splitting of a link in S3. The Kirby–Thompson invariant of a bridge trisection measures its complexity in terms of distances between disk sets in the pants complex of the trisection surface. We give the first significant bounds for the Kirby–Thompson invariant of spun knots. In particular, we show that the Kirby–Thompson invariant of the spun trefoil is 15. 
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