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Aug 13, 2020 at 21:28 comment added GageMartin I should also say, because of how "nice" representations of $U_q(\mathcal{sl}_2)$ are, there are deep relations between the colored Jones polynomial and Jones polynomials of cables. Sometimes the cabling construction is used as a definition of the colored Jones polynomial to make working with it easier
Aug 13, 2020 at 21:25 comment added GageMartin I have heard great things about "An Introduction to Quantum and Vassiliev Knot Invariants" by Jackson and Moffatt and at a glance it looks nice. I think Chapter 9 would have the details you are asking about
Aug 13, 2020 at 17:53 comment added Jake B. Thanks! Do you have any (readable) references for 1) The Jones obtained from the representation of $U_q(\mathfrak{sl}_2)$ and 2) The coloring/labeling of components?
Aug 6, 2020 at 17:52 history answered GageMartin CC BY-SA 4.0