# The Yang-Mills Measure in the $SU(3)$ Skein Module

## TL;DR

This paper constructs a local diffeomorphism invariant trace on the $SU(3)$-skein space of a surface cross an interval, extending the understanding of skein modules related to quantum invariants.

## Contribution

It introduces a new invariant trace on $SU(3)$-skein spaces for surfaces, providing a novel tool for studying 3-manifold invariants and quantum topology.

## Key findings

- Defined the $SU(3)$-skein space for 3-manifolds.
- Constructed a local diffeomorphism invariant trace on the skein space of surface products.
- Extended the framework for quantum invariants in $SU(3)$-skein theory.

## Abstract

Let $A\neq 0$ be a complex number with $ |A|\neq 1$. Let $M$ be a compact smooth oriented $3$-manifold, the $SU(3)$-skein space of $M$, $S_A(M)$, is the vector space over $\mathbb{C}$ generated by framed oriented links (including framed oriented trivalent graphs in $M$) quotient by the $SU(3)$-skein relations due to Kuperberg. For a closed, orientable surface $F$, we construct a local diffeomorphism invariant trace on $S_A(F\times I)$.