What type of study is this?

This is a Quantitative Study study.

October 16, 2025

Finiteness and Dimension of Stated Skein Modules over Frobenius

Key Points

The stated skein module is finitely generated over certain parameters, indicating structural properties.
An upper bound for the dimension of the skein module is provided when considering compact marked 3-manifolds.
The research reveals that the skein module captures important characteristics of punctured bordered surfaces.
The dimension of the module is closely tied to the boundary components and Euler characteristics of the surfaces considered.

Abstract

When the quantum parameter Formula: see text is a root of unity of odd order, the stated skein module Formula: see text has an Formula: see text-module structure, where Formula: see text is a marked 3-manifold. We prove Formula: see text is a finitely generated Formula: see text-module when Formula: see text is compact, which furthermore shows the reduced stated skein module for the compact marked 3-manifold is finite-dimensional over Formula: see text. We also give an upper bound for the dimension of Formula: see text over Formula: see text when Formula: see text is compact. For a punctured bordered surface Formula: see text, we use Formula: see text to denote the image of the Frobenius map when Formula: see text is a root of unity of odd order Formula: see text. Then Formula: see text lives in the center of the stated skein algebra Formula: see text. Let Formula: see text be the field of fractions of Formula: see text, and Formula: see text be Formula: see text. Then we show the dimension of Formula: see text over Formula: see text is Formula: see text where Formula: see text equals to the number of boundary components of Formula: see text minus the Euler characteristic of Formula: see text.

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