Relating Stated Skein Algebras and Internal Skein Algebras

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• ISSN: 1815-0659
• Main Author: Haïoun, Benjamin
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211626
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Relating Stated Skein Algebras and Internal Skein Algebras. Benjamin Haïoun. SIGMA 18 (2022), 042, 39 pages

We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges, and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edge internal skein algebras using excision properties of skein categories, and agree with excision properties of stated skein algebras when = q²(₂)-modᶠⁱⁿ. Our proofs are mostly based on skein theory, and we do not require the reader to be familiar with the formalism of higher categories.