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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. Chern–Simons functional, singular instantons, and the four-dimensional clasp number

   https://doi.org/10.4171/JEMS/1320  

   Daemi, Aliakbar ; Scaduto, Christopher ( April 2023 , Journal of the European Mathematical Society)

   Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern-Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Frøyshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless SU(2) representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided. 

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3. Big data approaches to knot theory: Understanding the structure of the Jones polynomial

   https://doi.org/10.1142/S021821652250095X  

   Levitt, Jesse S. ; Hajij, Mustafa ; Sazdanovic, Radmila ( November 2022 , Journal of Knot Theory and Its Ramifications)

   In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood. 

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4. Azumaya Algebras and Canonical Components

   https://doi.org/10.1093/imrn/rnaa209  

   Chinburg, Ted ; Reid, Alan W ; Stover, Matthew ( September 2020 , International Mathematics Research Notices)

   Abstract Let $M$ be a compact 3-manifold and $\Gamma =\pi _1(M)$. Work by Thurston and Culler–Shalen established the ${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$ character variety $X(\Gamma )$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $X(\Gamma )$, as well as distinguished points on the canonical component, when $\Gamma $ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields. 

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5. Azumaya algebras and canonical components

   Ted Chinburg, Alan Reid ( January 2022 , International mathematics research notices)

   Let M be a compact 3-manifold and 􏲣 = π1(M). The work by Thurston and Culler– Shalen established the SL2(C) character variety X(􏲣) as fundamental tool in the study of the geometry and topology of M. This is particularly the case when M is the exterior of a hyperbolic knot K in S3. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of X(􏲣), as well as distinguished points on the canonical component, when 􏲣 is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields. 

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