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1. The Strong Slope Conjecture and torus knot

   https://doi.org/10.2969/jmsj/81068106  

   Kalfagianni, Efstratia ( October 2020 , Journal of the Mathematical Society of Japan)

   null (Ed.)

   We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a $(2,q)$-torus knot. 

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2. Remarks on Jones slopes and surfaces of knots

   Kalfagianni, Efstratia ( May 2021 , Acta mathematica Vietnamica)

   null (Ed.)

   We point out that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure eight knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial. 

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3. Turaev-Viro invariants, colored Jones polynomial and volume.

   Renaud Detcherry, Efstratia Kalfagianni ( April 2018 , Quantum topology)

   We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem not to be predicted by neither the Kashaev-Murakami-Murakami volume conjecture and various of its generalizations nor by Zagier's quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify it for all knots with zero simplicial volume. Finally we observe that our simplicial volume conjecture is stable under connect sum and split unions of links. 

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4. Jones diameter and crossing numbers of knots

   Kalfagianni, E. ( February 2023 , Advances in Mathematics)

   It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite knots. More specifically, we exhibit minimal crossing number diagrams for untwisted Whitehead doubles of zero-writhe adequate knots. This allows us to determine the crossing number of untwisted Whitehead doubles of any amphicheiral adequate knot, including, for instance, the Whitehead doubles of the connected sum of any alternating knot with its mirror image. We also determine the crossing number of the connected sum of any adequate knot with an untwisted Whitehead double of a zero-writhe adequate knot. 

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5. Learning knot invariants across dimensions

   https://doi.org/10.21468/SciPostPhys.14.2.021  

   Craven, Jessica ; Hughes, Mark ; Jejjala, Vishnu ; Kar, Arjun ( January 2023 , SciPost Physics)

   We use deep neural networks to machine learn correlations betweenknot invariants in various dimensions. The three-dimensional invariantof interest is the Jones polynomial J(q) J ( q ) ,and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t) Kh ( q , t ) ,smooth slice genus g g ,and Rasmussen’s s s -invariant.We find that a two-layer feed-forward neural network can predict s s from \text{Kh}(q,-q^{-4}) Kh ( q , − q − 4 ) with greater than 99% 99 % accuracy. A theoretical explanation for this performance exists in knottheory via the now disproven knight move conjecture, which is obeyed byall knots in our dataset. More surprisingly, we find similar performancefor the prediction of s s from \text{Kh}(q,-q^{-2}) Kh ( q , − q − 2 ) ,which suggests a novel relationship between the Khovanov and Leehomology theories of a knot. The network predicts g g from \text{Kh}(q,t) Kh ( q , t ) with similarly high accuracy, and we discuss the extent to which themachine is learning s s as opposed to g g ,since there is a general inequality |s| ≤2g | s | ≤ 2 g .The Jones polynomial, as a three-dimensional invariant, is not obviouslyrelated to s s or g g ,but the network achieves greater than 95% 95 % accuracy in predicting either from J(q) J ( q ) .Moreover, similar accuracy can be achieved by evaluating J(q) J ( q ) at roots of unity. This suggests a relationship with SU(2) S U ( 2 ) Chern—Simons theory, and we review the gauge theory construction ofKhovanov homology which may be relevant for explaining the network’sperformance. 

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