A comparison between $SL_n$ spider categories

TL;DR

This paper proves a conjecture by comparing different $SL_n$ skein theories, showing their equivalence as categories and isomorphism of associated skein modules, thus unifying various approaches in the field.

Contribution

It establishes the equivalence of the Cautis-Kamnitzer-Morrison subcategory with Sikora's quotient category and shows their skein modules are isomorphic, confirming a conjecture and answering a Ph.D. thesis question.

Findings

Equivalence of the CKM subcategory and Sikora's quotient category as spherical braided categories.

Isomorphism of skein modules associated to CKM and Sikora's webs.

Confirmation of a conjecture by Lê and Sikora.

Abstract

We prove a conjecture of L\^{e} and Sikora by providing a comparison between various existing $SL_n$ skein theories. While doing so, we show that the full subcategory of the spider category, $\mathcal{S}p(SL_n)$, 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.