  The definition of the Jones polynomial in the 80's gave rise to a large
family of so-called quantum link invariants, based on quantum groups. These
quantum invariants are all controlled by the same two-variable invariant (the
HOMFLY-PT polynomial), which also specializes to the older Alexander
polynomial. Building upon quantum Schur--Weyl duality and variants of this
phenomenon, I will explain an algebraic setup that allows for global
definitions of these quantum polynomials, and discuss extensions of these
quantum objects designed to encompass all of the mentioned invariants,
including the HOMFLY-PT polynomial.
