Hiroaki Karuo

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.

We introduce a joint generalization, called LRY skein algebras, of Kauffman bracket skein algebras (of surfaces) that encompasses both Roger-Yang skein algebras and stated skein algebras. We will show that, over an arbitrary ground ring which is a commutative domain, the LRY skein algebras are domains and have degenerations (by filtrations) equal to monomial subalgebras of quantum tori. This integrality answers a question of Roger-Yang for the most general ground ring. We also calculate the Gelfand-Kirillov dimension of LRY algebras and show they are Noetherian if the ground ring is. Moreover they are orderly finitely generated. To study the LRY algebras and prove the above-mentioned results, we construct quantum traces, both the so-called X-version for all surfaces and also an A-version for a smaller class of surfaces. We also introduce a modified version of Dehn-Thurston coordinates for curves which are more suitable for the study of skein algebras as they pick up the highest degree terms of products in certain natural filtrations.

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.