Unstable Vassiliev Theory

TL;DR

This paper develops an inverse system of spectral sequences on spaces of plumbers' knots to model the homotopy type of long knots, connecting finite type invariants with the limit of these sequences.

Contribution

It introduces a novel inverse system of unstable Vassiliev spectral sequences on plumbers' knots and extends Vassiliev derivatives to all singularity types.

Findings

The limit of the spectral sequences contains finite type invariants.

Constructs a cell structure on the discriminant of plumbers' curves.

Extends Vassiliev derivatives to all singularity types.

Abstract

We construct an inverse system of unstable Vassiliev spectral sequences on the spaces of plumbers' knots, which model the homotopy type of the space of long knots, and show that the limit of these sequences contains the finite type invariants in their usual complexity. Utilizing the cell structure on the discriminant of the spaces of plumbers curves, we extend the notion of Vassiliev derivative to all singularity types of plumbers' curves.