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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. Slope of orderable Dehn filling of two-bridge knots

   https://doi.org/10.1142/S0218216522500067  

   Gao, Xinghua (January 2022, Journal of Knot Theory and Its Ramifications)

   In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope [Formula: see text] such that [Formula: see text]-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots. 

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5. 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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