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1. 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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2. Rank bounds in link Floer homology and detection results

   https://doi.org/10.4171/QT/197  

   Binns, Fraser; Dey, Subhankar (November 2023, Quantum Topology)

   Viewing the BRAID invariant as a generator of link Floer homology, we generalize work of Baldwin–Vela-Vick to obtain rank bounds on the next-to-top grading of knot Floer homology. These allow us to classify links with knot Floer homology of rank at most eight and prove a variant of a classification of links with Khovanov homology of low rank due to Xie–Zhang. In another direction, we use a variant of Ozsváth–Szabó's classification ofE_2collapsed\mathbb{Z}\oplus\mathbb{Z}filtered chain complexes to show that knot Floer homology detectsT(2,8)andT(2,10). Combining these techniques with the spectral sequences of Batson–Seed, Dowlin, and Lee, we can show that Khovanov homology likewise detectsT(2,8)andT(2,10). 

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3. Meridional rank and bridge number of knotted 2-spheres

   https://doi.org/10.4153/S0008414X23000883  

   Joseph, Jason; Pongtanapaisan, Puttipong (February 2025, Canadian Journal of Mathematics)

   Abstract The meridional rank conjecture asks whether the bridge number of a knot in$$S^3$$is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in$$S^4$$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres. 

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4. Sutured instanton homology and Heegaard diagrams

   https://doi.org/10.1112/S0010437X23007303  

   Baldwin, John A; Li, Zhenkun; Ye, Fan (September 2023, Compositio Mathematica)

   Suppose$$\mathcal {H}$$is an admissible Heegaard diagram for a balanced sutured manifold$$(M,\gamma )$$. We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology$$\mathit {SHI}(M,\gamma )$$. It follows, in particular, that strong L-spaces are instanton L-spaces. 

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5. Murasugi sum and extremal knot Floer homology

   https://doi.org/10.4171/qt/186  

   Cheng, Zhechi; Hedden, Matthew; Sarkar, Sucharit (March 2024, Quantum Topology)

   The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni’s isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that\tau=gfor each summand if and only if\tau=gholds for the Murasugi sum (with\tauandgdefined appropriately for multi-component links). Some applications are presented. 

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