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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. A Kirby color for Khovanov homology

   https://doi.org/10.4171/JEMS/1589  

   Hogancamp, Matthew; Rose, David_E V; Wedrich, Paul (February 2025, Journal of the European Mathematical Society)

   We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu–Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the\mathfrak{gl}_{2}skein lasagna module, hence is an invariant of 4-dimensional 2-handlebodies. 

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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. Doubly pointed trisection diagrams and surgery on 2-knots

   https://doi.org/10.1017/S0305004121000165  

   GAY, DAVID; MEIER, JEFFREY (January 2022, Mathematical Proceedings of the Cambridge Philosophical Society)

   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. 

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5. Reflections on trisection genus

   Chu, Michelle; Tillmann, Stephan (January 2019, Revue roumaine de mathématiques pures et appliquées)

   Fukui, Toshizumi; Koike, Satoshi; Paunescu, Laurențiu (Ed.)

   The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that analogous results concerning the trisection genus of a smooth, orientable 4-manifold have more general answers than their counterparts for 3-manifolds. In the case of hyperbolic 4-manifolds, upper and lower bounds are given in terms of volume and a trisection of the Davis manifold is described. 

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