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1. Khovanov Laplacian and Khovanov Dirac for knots and links

   https://doi.org/10.1088/2632-072X/adde9f  

   Jones, Benjamin; Wei, Guo-Wei (June 2025, Journal of Physics: Complexity)

   Abstract Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology. 

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2. Gluing maps and cobordism maps for sutured monopole Floer homology.

   Li, Zhenkun (October 2018, ArXiv.org)

   In this paper we construct gluing maps and cobordism maps for sutured monopole Floer homology. 

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3. PL-Genus of surfaces in homology balls

   https://doi.org/10.1017/fms.2023.126  

   Hom, Jennifer; Stoffregen, Matthew; Zhou, Hugo (January 2024, Forum of Mathematics, Sigma)

   Abstract We consider manifold-knot pairs$$(Y,K)$$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$$\Sigma $$in a homology ballX, such that$$\partial (X, \Sigma ) = (Y, K)$$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$$(Y, K)$$to any knot in$$S^3$$can be arbitrarily large. The proof relies on Heegaard Floer homology. 

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4. 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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5. 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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