Algebraic structures of Vassiliev invariants for knot families

TL;DR

This paper investigates the algebraic structure of Vassiliev knot invariants within parametric knot families, proposing conjectures on their finite generation and independence, and exploring their role as complete invariants.

Contribution

It introduces conjectures on the finite generation of Vassiliev invariants in 1-parametric families and analyzes algebraic independence in multi-parametric families.

Findings

In 1-parametric families, Vassiliev invariants are conjectured to be finitely generated.

In multi-parametric families, examples show an infinite number of generators.

Number of algebraically independent invariants equals the number of parameters in studied examples.

Abstract

We explore algebraic relations on Vassiliev knot invariants related with correlators in the 3-dimensional Chern--Simons theory. Vassiliev invariants form infinite-dimensional algebra. We focus on $k$-parametric knot families with Vassiliev invariants being polynomials in family parameters. We conjecture that such 1-parametric algebra of Vassiliev invariants is always finitely generated, while in the case of more parameters, we provide example of the knot family with infinite number of generators. Inside a knot family, there appear extra algebraic relations on Vassiliev invariants. We show that there are $\leq k$ algebraically independent Vassiliev invariants for $k$-parametric knot family. However, in all our examples, the number of algebraically independent Vassiliev invariants is exactly $k$, and it is open question if there exists a $k$-parametric knot family with a fewer number of algebraically independent Vassiliev invariants. We also demonstrate that a complete knot invariant of some $k$-parametric knot families consists of $k$ Vassiliev invariants.