Skein modules, character varieties and essential surfaces of 3-manifolds

The $SL_2(C)$-skein modules of closed 3-manifolds were defined by in the 90’s but till recently little was known about their structure. The modules depend on a parameter A and can be considered over $ {\mathbb Z}[A^{\pm 1}]$ or over ${\mathbb Q}(A)$.

The ${\mathbb Q}(A)$-module is known to be finitely generated  while the structure over ${\mathbb Z}[A^{\pm 1}]$ can be complicated.

We will discuss how the existence of "essential" surfaces in manifolds reflects on the structure of their $ {\mathbb Z}[A^{\pm 1}]$- module.  We will also discuss how this information allows to compute the dimension of the ${\mathbb Q}(A)$- modules for "small” manifolds,  and understand their  relation to their character varieties.