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1. 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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2. The slope conjecture for Montesinos knots

   https://doi.org/10.1142/S0129167X20500561  

   Garoufalidis, Stavros; Lee, Christine Ruey; van der Veen, Roland (June 2020, International Journal of Mathematics)

   The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions. 

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3. Periodic partial theta functions and q-hypergeometric knot multisums as quantum Jacobi forms

   https://doi.org/10.1016/j.jmaa.2023.127727  

   Folsom, Amanda (February 2024, Journal of Mathematical Analysis and Applications)

   We prove that general two-variable partial theta functions with periodic coefficients are quantum Jacobi forms, and establish their explicit transformation and analytic properties. As applications, we also prove that seven infinite families of q-hypergeometric multisums and related partial theta functions of interest arising from certain knot colored Jones polynomials, Kashaev invariants for torus knots and Virasoro characters, and “strange” identities, appearing in (separate) works of Bijaoui et al., Hikami, Hikami-Kirillov, Lovejoy, and Zagier are quantum Jacobi forms. 

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4. 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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5. Crossing numbers of cable knots

   https://doi.org/10.1112/blms.13140  

   Kalfagianni, Efstratia; Mcconkey, Rob (June 2024, Bulletin of the London Mathematical Society)

   We use the degree of the colored Jones knot polynomials to show that the crossing number of a (p,q)‐cable of an adequate knot with crossing number c is larger than q^2 c. As an application, we determine the crossing number of 2‐cables of adequate knots. We also determine the crossing number of the connected sum of any adequate knot with a 2‐cable of an adequate knot. 

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