Spider evaluation and representations of web groups

Abstract

The topology of [Formula: see text]-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to [Formula: see text] 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 [Formula: see text] from the root of the tree to the tip of a leaf an irreducible component [Formula: see text] of the representation variety of the web, and a graded subalgebra [Formula: see text] of [Formula: see text]. The spider evaluation of geodesic [Formula: see text] is the symmetrized Poincaré polynomial of [Formula: see text]. The spider evaluation of the web is the sum of the symmetrized Poincaré polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves.