# Virtual classes of parabolic $\mathrm{SL}_2(\mathbb{C})$-character varieties

## TL;DR

This paper computes the virtual classes of SL_2(C)-character varieties of surfaces with parabolic points, revealing new recursive interaction patterns when punctures are semi-simple non-generic.

## Contribution

It introduces a method to compute virtual classes for these character varieties and uncovers a novel recursive pattern in the semi-simple non-generic case.

## Key findings

- Identification of recursive interaction patterns
- Explicit computation of virtual classes
- New phenomena in semi-simple non-generic cases

## Abstract

In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are chosen to be semi-simple non-generic, we show that a new interaction phenomenon appears generating a recursive pattern.