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1. 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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2. Guts, Volume and skein modules of 3-manifolds

   Bavier, Brandon; Kalfagianni, Efstratia (June 2021, Fundamenta Mathematicae)

   null (Ed.)

   We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link. 

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3. Residual torsion-free nilpotence, bi-orderability, and two-bridge links

   https://doi.org/10.4153/S0008414X2300007X  

   Johnson, Jonathan (January 2023, Canadian Journal of Mathematics)

   Abstract Residual torsion-free nilpotence has proved to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of para-free groups. This paper proves Mayland’s assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $$\mathbb{Z}$$ . We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link groups. This proof makes use of a modified version of a graph-theoretic construction of Hirasawa and Murasugi in order to understand the structure of the Alexander subgroup for a two-bridge link group. 

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4. Log-Concavity of the Alexander Polynomial

   https://doi.org/10.1093/imrn/rnae058  

   Hafner, Elena S; Mészáros, Karola; Vidinas, Alexander (April 2024, International Mathematics Research Notices)

   Abstract The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $$\Delta _{L}(t)$$ of an alternating link $$L$$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus $$2$$ alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $$\Delta _{L}(t)$$, where $$L$$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. 

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5. OOD Link Prediction Generalization Capabilities of Message-Passing GNNs in Larger Test Graphs

   Zhou, Yangze; Kutyniok, Gitta; Ribeiro, Bruno (December 2022, Advances in neural information processing systems)

   This work provides the first theoretical study on the ability of graph Message Passing Neural Networks (gMPNNs) -- such as Graph Neural Networks (GNNs) -- to perform inductive out-of-distribution (OOD) link prediction tasks, where deployment (test) graph sizes are larger than training graphs. We first prove non-asymptotic bounds showing that link predictors based on permutation-equivariant (structural) node embeddings obtained by gMPNNs can converge to a random guess as test graphs get larger. We then propose a theoretically-sound gMPNN that outputs structural pairwise (2-node) embeddings and prove non-asymptotic bounds showing that, as test graphs grow, these embeddings converge to embeddings of a continuous function that retains its ability to predict links OOD. Empirical results on random graphs show agreement with our theoretical results. 

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