Producing knot/link invariants is "as simple as" finding functions on the braid group invariant under the Markov moves. Many classical invariants arise from the character of a representation of the braid group: this is guaranteed to be constant on conjugacy classes (so it is fixed by one class of Markov moves), and for reps satisfying some extra properties we can build an invariant that is fully Markov move invariant -- see for example ch15 of Chari and Pressley's book on quantum groups.
Since there is a surjective map from the braid group to the symmetric group, every symmetric group representation gives rise to a representation of the braid group. My question is: can we build interesting knot invariants from symmetric group representations? I guess not, but I hope someone can explain why.