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Optimistic limit of the colored Jones polynomial and the existence of a solution


Author: Jinseok Cho
Journal: Proc. Amer. Math. Soc. 144 (2016), 1803-1814
MSC (2010): Primary 57M50; Secondary 57M27, 51M25, 58J28
DOI: https://doi.org/10.1090/proc/12845
Published electronically: September 9, 2015
MathSciNet review: 3451255
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Abstract: For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic representation $ \rho :\pi _1(L)\rightarrow {\rm PSL}(2,\mathbb{C})$. This gives us a very simple and combinatorial method to calculate the complex volume of $ \rho $.


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Additional Information

Jinseok Cho
Affiliation: Pohang Mathematics Institute (PMI), Pohang 790-784, Republic of Korea
Email: dol0425@gmail.com

DOI: https://doi.org/10.1090/proc/12845
Received by editor(s): October 16, 2014
Received by editor(s) in revised form: April 28, 2015
Published electronically: September 9, 2015
Additional Notes: The author thanks Seonhwa Kim, who had already predicted the existence of a solution when the author defined the potential function of the colored Jones polynomial several years ago. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014047764) and NRF-2015R1C1A1A02037540.
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2015 American Mathematical Society