I want to distinguish between links where the components have different (or same) colors. In the Alexander polynomial we can assign a different variable to each component, but what about a Kauffman bracket? Is there a generalization so that we could distinguish between multi-component links where some of the components are colored?
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The Jones polynomial can be understood as coming from representations of quantum groups. The regular Jones polynomial comes from the defining representation of $U_q(\mathfrak{sl}_2)$. It is possible to color/label components of the link with different representations of $U_q(\mathfrak{sl}_2)$ to obtain various colored Jones polynomials.
However, this is a little different than the multivariable Alexander polynomial because the invariant you end up with is still a polynomial in a single variable.
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$\begingroup$ 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? $\endgroup$– Jake B.Commented Aug 13, 2020 at 17:53
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$\begingroup$ 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 $\endgroup$ Commented Aug 13, 2020 at 21:25
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$\begingroup$ 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 $\endgroup$ Commented Aug 13, 2020 at 21:28