The Schur-Weyl Duality is a classical result that relates the representation theory of \mathfrac{sl}_2 and the symmetric group via their action on tensor powers of \C^2. We also have a result due to Weyl Rummer and Teller that gives a diagrammatic interpretation of the\mathfrac{sl}_2 invariants of (\C^2)^{2n}. We shall see these results and their corresponding generalizations to $U_q{\mathfrac{sl}_2)$ on one hand and the Hecke algebra on one hand. We will also make surprising connections to Link Invariants and see how our diagrammatic calculus gives us the Jones polynomial.