UC Santa Barbara

Skein modules of 3-manifolds are situated at the intersection of low-dimensional topology and representation theory. The skein module of a thickened surface has a natural algebra structure induced by the superposition of skeins. In this thesis, we study connections between quantum groups and these skein algebras by focusing on the SL_3 skein algebra of an oriented punctured surface \Sigma, which may have punctured boundary components. Generalizing a construction of L\^{e}, we associate an SL_3 stated skein algebra to any such \Sigma. These algebras admit natural algebra morphisms, called splitting maps, associated to the splitting of surfaces along ideal arcs. We give an explicit basis for the SL_3 stated skein algebra, which is an extension of the Sikora-Westbury basis for the ordinary SL_3 skein algebra. Using this basis, we show that the splitting maps are injective and describe their images. Applying the splitting maps to a triangulable surface, we obtain a triangular decomposition in which we embed the skein algebra in a domain that has an explicit presentation described in terms of the quantum group O_q(SL_3). The ingredients we collect along the way allow for a skein-theoretic method of recovering the fact that Kuperberg's webs describe a full subcategory of the representation category of U_q(sl_3).