Hiroaki Karuo

We consider a generalization of the Kauffman bracket skein algebra of a surface that is generated by loops and arcs between marked points on the interior or boundary, up to skein relations defined by Muller and Roger-Yang. We compute the center of this Muller-Roger-Yang skein algebra and show that it is almost Azumaya when the quantum parameter $q$ is a primitive $n$-th root of unity with odd $n$. We also discuss the implications on the representation theory of the Muller-Roger-Yang generalized skein algebra.

We study a generalized Witten's finiteness conjecture for the skein modules of oriented compact 3-manifolds with boundary. We formulate an equivalent version of the generalized finiteness conjecture using handlebodies and 2-handles, and prove the conjecture for some classes with the handlebodies of genus 2 and 3 using the equivalent version.

Abstract We generalize the quantum duality map $\mathbb {I}_{\mathcal {A } }$ of Allegretti–Kim (Adv Math 306:1164–1208, 2017) for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map $\mathbb {I}_{\mathcal {X } }$ on the dual side based on the quantum bracelets basis (Mandel and Qin in arXiv:2301.11101; Thurston in Proc Natl Acad Sci USA 111(27):9725–9732, 2014). Our construction factors through reduced stated skein algebras, based on the quantum trace maps (Lê in Quantum Topol 9(3):591–632, 2018) together with an appropriate way of skein lifting of integral $\mathcal {A}$-laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.

In the paper, we show some properties of (reduced) stated SL($n$)-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated SL($n$)-skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can understand the representation theory of (reduced) stated SL($n$)-skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access to the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially.