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1. Bridge trisections of knotted surfaces in 4-manifolds

   https://doi.org/10.1073/pnas.1717171115  

   Meier, Jeffrey; Zupan, Alexander (October 2018, Proceedings of the National Academy of Sciences)

   We prove that every smoothly embedded surface in a 4-manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4-manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a generalized bridge trisection, extends the authors’ definition of bridge trisections for surfaces in S 4 . Using this construction, we give diagrammatic representations called shadow diagrams for knotted surfaces in 4-manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside ℂ ℙ 2 . Using these examples, we prove that there exist exotic 4-manifolds with ( g , 0 ) —trisections for certain values of g. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams. 

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2. Computing a Link Diagram from its Exterior

   Dunfield, Nathan M.; Obeidin, Malik; Rudd, Cameron Gates (January 2022, Leibniz international proceedings in informatics)

   Goaoc, Xavier; Kerber, Michael (Ed.)

   A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture. 

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3. Trisections and link surgeries

   https://doi.org/10.53733/94  

   Kirby, Robion; Thompson, Abigail (September 2021, New Zealand Journal of Mathematics)

   We examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby \cite{G-K3}. These links lie on Heegaard surfaces in $$\#^j S^1 \times S^2$$ and have surgeries yielding $$\#^k S^1 \times S^2$$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer. 

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4. Bounding the Kirby–Thompson invariant of spun knots

   https://doi.org/10.2140/agt.2024.24.3363  

   Aranda, Román; Pongtanapaisan, Puttipong; Taylor, Scott A; Zhang, Suixin Cindy (January 2024, 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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5. Floer homology and non-fibered knot detection

   https://doi.org/10.1017/fmp.2024.28  

   Baldwin, John A; Sivek, Steven (January 2025, Forum of Mathematics, Pi)

   Abstract We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots and that HOMFLY homology detects infinitely many knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that$$0$$-surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R Conjecture. 

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