A Representation Theorem for Knots and a Generalization of the Fundamental Theorem of Finite Type Invariants

TL;DR

This paper introduces a method to generate all knots in three-dimensional space from signed chord diagrams, generalizes a key theorem for finite type invariants, and establishes moves connecting different diagrams of the same knot.

Contribution

It presents a new representation theorem linking knots and signed chord diagrams and extends the fundamental theorem of finite type invariants.

Findings

Every knot can be represented by signed chord diagrams.

The paper generalizes the fundamental theorem of finite type invariants.

Moves are provided to connect different diagrams of the same knot.

Abstract

We provide a way to produce knots in $S^3$ from signed chord diagrams, and prove that every knot can be produced in this way. Using these diagrams, we generalize the fundamental theorem of finite type invariants. We also provide moves for the diagrams so that any two diagrams for the same knot are connected by a sequence of moves.