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We study various specializations of the colored HOMFLY-PT polynomial. Thesespecializations are used to show that the multivariable link invariants arisingfrom a complex family of sl(m|n) super-modules previously defined by theauthors contains both the multivariable Alexander polynomial and Kashaev'sinvariants. We conjecture these multivariable link invariants also specializeto the generalized multivariable Alexander invariants defined by Y. Akutsu, T.Deguchi, and T. Ohtsuki.

On the Colored HOMFLY-PT, Multivariable and Kashaev Link Invariants