Abstract: Let V be an r-dimensional vector space over the complex numbers and let
V* be its dual space. The special linear group SL(V) acts naturally on
the coordinate ring of V^n x V*^m, which is a polynomial ring in r(n+m) variables. We are interested in polynomials which are invariant under this action. In particular we are interested in a basis of the invariant space in terms of planar diagrams that exhibits lot of the symmetries of the invariant space. In 1996 Kuperberg introduced such a diagrammatic basis
for r=3, known as SL_3 web basis. Although spanning sets and relations are known for arbitrary r, it remained unclear how to extract a basis for r>3. In joint work with Christian Gaetz, Oliver Pechenik, Jessica Striker and Joshua Swanson we are able to construct a web basis for SL_4. To do so, we give a crystal theoretic perspective on Kuperberg's work, which allows generalisation.