Representations of knot groups in ${\text{AGL}}_1(\mathbb{C})$ and Alexander invariants

Abstract

This paper reinterprets Alexander-type invariants of knots via representation varieties of knot groups into the group ${\text{AGL}}_1(\mathbb{C})$ of affine transformations of the complex line. In particular, we prove that the coordinate ring of the ${\text{AGL}}_1(\mathbb{C})$-representation variety is isomorphic to the symmetric algebra of the Alexander module. This yields a natural interpretation of the Alexander polynomial as the singular locus of a coherent sheaf over ${\mathbb{C}}^{\ast }$, whose fibres correspond to quandle representation varieties of the knot quandle. As a by-product, we construct Topological Quantum Field Theories that provide effective computational methods and recover the Burau representations of braids. This theory offers a new geometric perspective on classical Alexander invariants and their functorial quantization.

34 pages, 10 figures