A comparison between $SL_n$ spider categories

Anup Poudel

A comparison between $SL_n$ spider categories

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arxiv 2210.09289 v5 pith:TNIE6CSA submitted 2022-10-17 math.GT math.QA

Anup Poudel This is my paper · ORCID

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keywords categorysikoracomparisonskeinspideranswersassociatedbraided

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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.

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Naturality of ${\rm SL}_n$ quantum trace maps for surfaces
math.QA 2024-12 conditional novelty 7.0 of 10

For every n, the SL_n quantum trace maps for different ideal triangulations are related by a balanced n-th root quantum coordinate change that extends the Fock-Goncharov isomorphism.

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