Algebraic structures of Vassiliev invariants for knot families

TL;DR

The paper investigates how Vassiliev knot invariants, as correlators in 3d Chern–Simons theory, organize within k-parameter knot families whose invariants are polynomial in the parameters. It proves a general bound: a k-parameter family has at most k algebraically independent Vassiliev invariants, and it provides explicit examples showing both finitely generated (1-parameter) and infinitely generated (multi-parameter) algebras. Through detailed analyses of families such as 2-strand torus, twist, pretzel, Kanenobu, and Stanford knots, the authors reveal pervasive algebraic relations among invariants, and demonstrate cases where a complete invariant is realized by a small set of invariants. The work highlights that while some families admit a complete invariant consisting of a handful of Vassiliev invariants, others (notably torus knot families T[m,n]) yield infinitely many primaries, underscoring rich and intricate algebraic structures in knot theory tied to CS theory. These results advance understanding of how knot topology constrains perturbative invariants and suggest directions for identifying minimal generating sets and complete invariants within knot families.

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.

Figures (8)

• Figure 1: A double point.
• Figure 2: Vassiliev skein relation.
• Figure 3: By definition, the zeroth Vassiliev invariant vanishes on all singular knots and, in particular, on knots with one double point. Being applied to torus knots together with Vassiliev skein relation, this leads to the fact that the zeroth Vassiliev invariant is a constant.
• Figure 4: In the first row we write the condition that the first Vassiliev invariant turns to zero on a torus knot with two double points. Then, we resolve the upper double point with the use of Vassiliev skein relation in Fig. \ref{['fig:Vass-skein']}. In the second line, we apply Vassiliev skein relation to the remaining double point and obtain the difference relation that restricts the first Vassiliev invariant for a torus knot $T[2,n]$ to be linear in $n$.
• Figure 5: Twist knot Tw$_{2k}$.

Theorems & Definitions (4)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4