A comparison between $S{L}_n$ spider categories

Abstract

We prove a conjecture of Lê and Sikora by providing a comparison between various existing $S{L}_n$ skein theories. While doing so, we show that the full subcategory of the spider category, $\mathcal{S}p(S{L}_n)$, defined by Cautis-Kamnitzer-Morrison, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This also answers a question from Morrison's Ph.D. thesis. Finally, we show that the skein modules associated to the CKM and Sikora's webs are isomorphic.

30 pages, some changes made to the structure of the paper. To appear in Canadian Journal of Math