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1. Khovanov homology and exotic surfaces in the 4-ball

   https://doi.org/10.1515/crelle-2024-0001  

   Hayden, Kyle; Sundberg, Isaac (March 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms).We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations.We also show that Plamenevskaya’s transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms. 

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2. Khovanov homology and the cinquefoil

   https://doi.org/10.4171/JEMS/1415  

   Baldwin, John A; Hu, Ying; Sivek, Steven (April 2025, Journal of the European Mathematical Society)

   We prove that Khovanov homology with coefficients in\Z/2\Zdetects the(2,5)torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy. We combine these tools with classical results on the dynamics of surface homeomorphisms to reduce the detection question to a problem about mutually braided unknots, which we then solve with computer assistance. 

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3. Extremal Khovanov homology and the girth of a knot

   https://doi.org/10.1142/S0218216522500833  

   Sazdanović, Radmila; Scofield, Daniel (November 2022, Journal of Knot Theory and Its Ramifications)

   We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to [Formula: see text], then the girth of the link is equal to [Formula: see text] 

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4. A mixed invariant of nonorientable surfaces in equivariant Khovanov homology

   https://doi.org/10.1090/tran/8736  

   Lipshitz, Robert; Sarkar, Sucharit (December 2022, Transactions of the American Mathematical Society)

   We construct a mixed invariant of nonorientable surfaces from the Lee and Bar-Natan deformations of Khovanov homology and use it to distinguish pairs of surfaces bounded by the same knot, including some exotic examples. 

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5. 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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