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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.

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up vote 7 down vote accepted

The only information the symmetric group quotient keeps about an element of a braid group is how strings connect vertices. The only information about the corresponding link you can determine from that is the total number of components (equal to the number of cycles of the corresponding permutation). All other information is destroyed by the relations $s_i^2 = 1$.

However (and you probably know this, but it's worth saying) instead of mapping into the symmetric group, you can map into the Hecke algebra of $S_n$ (so instead quotient by a relation like $s_i^2 = (q - 1) s_i + 1$). The representation theory of the Hecke algebra is (most of the time) just like the representation theory of $S_n$, but you can actually get interesting invariants this way. See, for example, Kassel and Turaev's Braid groups.

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