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1. The ϒ function of L –space knots is a Legendre transform

   https://doi.org/10.1017/S030500411700024X  

   BORODZIK, MACIEJ; HEDDEN, MATTHEW (May 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   null (Ed.)

   Abstract Given an L –space knot we show that its ϒ function is the Legendre transform of a counting function equivalent to the d –invariants of its large surgeries. The unknotting obstruction obtained for the ϒ function is, in the case of L –space knots, contained in the d –invariants of large surgeries. Generalisations apply for connected sums of L –space knots, which imply that the slice obstruction provided by ϒ on the subgroup of concordance generated by L –space knots is no finer than that provided by the d –invariants. 

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2. From zero surgeries to candidates for exotic definite 4‐manifolds

   https://doi.org/10.1112/jlms.12800  

   Manolescu, Ciprian; Piccirillo, Lisa (August 2023, Journal of the London Mathematical Society)

   Abstract One strategy for distinguishing smooth structures on closed 4‐manifolds is to produce a knot in that is slice in one smooth filling of but not slice in some homeomorphic smooth filling . In this paper, we explore how 0‐surgery homeomorphisms can be used to potentially construct exotic pairs of this form. To systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find five topologically slice knots such that, if any of them were slice, we would obtain an exotic 4‐sphere. We also investigate the possibility of constructing exotic smooth structures on in a similar fashion. 

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3. Heegaard Floer homology and cosmetic surgeries in $S^3$

   https://doi.org/10.4171/JEMS/1218  

   Hanselman, Jonathan (March 2022, Journal of the European Mathematical Society)

   If a knot K in S^3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either ±2 or ±1/q for some value of q that is explicitly determined by the knot Floer homology of K. Moreover, in the former case the genus of K must be 2, and in the latter case there is a bound relating q to the genus and the Heegaard Floer thickness of K. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to 2. We also show that the conjecture holds for any knot K for which each prime summand of K has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes ±1 and ±2 on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access. 

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4. On the tau invariants in instanton and monopole Floer theories

   https://doi.org/10.1112/topo.12346  

   Ghosh, Sudipta; Li, Zhenkun; Wong, C‐M Michael (June 2024, Journal of Topology)

   Abstract We unify two existing approaches to thetauinvariants in instanton and monopole Floer theories, by identifying , defined by the second author via theminusflavors and of the knot homologies, with , defined by Baldwin and Sivek via cobordism maps of the 3‐manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute and for twist knots. 

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5. Going To The Other Side Via The Resurgent Bridge

   Costin, O; Dunne, G; Gruen, A; Gukov, S (October 2023, arXiv)

   arXiv (Ed.)

   Using resurgent analysis we offer a novel mathematical perspective on a curious bijection (duality) that has many potential applications ranging from the theory of vertex algebras to the physics of SCFTs in various dimensions, to q-series invariants in low-dimensional topology that arise e.g. in Vafa-Witten theory and in non-perturbative completion of complex Chern-Simons theory. In particular, we introduce explicit numerical algorithms that efficiently implement this bijection. This bijection is founded on preservation of relations, a fundamental property of resurgent functions. Using resurgent analysis we find new structures and patterns in complex Chern-Simons theory on closed hyperbolic 3-manifolds obtained by surgeries on hyperbolic twist knots. The Borel plane exhibits several intriguing hints of a new form of integrability. An important role in this analysis is played by the twisted Alexander polynomials and the adjoint Reidemeister torsion, which help us determine the Stokes data. The method of singularity elimination enables extraction of geometric data even for very distant Borel singularities, leading to detailed non-perturbative information from perturbative data. We also introduce a new double-scaling limit to probe 0-surgeries from the limiting r → ∞ behavior of 1 r surgeries, and apply it to the family of hyperbolic twist knots. 

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