# Spider Evaluation and Representations of Web Groups

## TL;DR

This paper investigates the topology of $SU(3)$-representation varieties of planar web fundamental groups, linking their structure to spider evaluations via algebraic and geometric methods.

## Contribution

It introduces a novel connection between web evaluation via spiders and the topology of associated representation varieties, using geodesics and cohomological invariants.

## Key findings

- Representation varieties are linked to spider evaluations.
- Each geodesic corresponds to an irreducible component and a cohomology algebra.
- Spider evaluation equals the sum of symmetrized Poincare polynomials.

## Abstract

The topology of $SU(3)$-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to $-1$ are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic $\gamma$ from the root of the tree to the tip of a leaf an irreducible component $C_{\gamma}$ of the representation variety of the web, and a graded subalgebra $A_{\gamma}$ of $H^*(C_{\gamma};\mathbb{Q})$. The spider evaluation of geodesic $\gamma$ is the symmetrized Poincare polynomial of $A_{\gamma}$. The spider evaluation of the web is the sum of the symmetrized Poincare polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves