bibkey: laninasleptsov2026vassilievfamilies
authors: E. Lanina; A. Sleptsov
year: 2026
title: "Algebraic structures of Vassiliev invariants for knot families"
doi: 10.1016/j.nuclphysb.2025.117268
url: https://arxiv.org/abs/2508.02385v2
claim: "The paper studies rational Vassiliev invariants restricted to knot families on which they are polynomials in the family parameters, shows that a k-parameter family has at most k algebraically independent such invariants, gives a two-parameter family whose algebra of invariants is infinitely generated, and conjectures that the algebra of polynomial Vassiliev invariants of every one-parameter family is finitely generated."
strata_touched:
  - D5/S3/Topology/KnotFamilyVassilievFiniteGeneration
license: citation-only
triage: anchor

Algebraic structures of Vassiliev invariants for knot families

E. Lanina and A. Sleptsov, arXiv:2508.02385v2 (journal version; v1 August 2025); Nuclear Physics B 1022 (2026), 117268. Quotations are from the v2 TeX source, with citations and equation references omitted.

The values and the algebra:

In this paper, we only deal with the Vassiliev invariants coming from the Chern–Simons correlation functions which are just rational numbers.

The set of all Vassiliev invariants forms commutative filtered graded algebra with respect to the usual multiplication of functions.

Primary invariants of a family:

If now, we descend from the set of all knots to a knot family, additional relations on primitive Vassiliev invariants can appear, and their subset will generate the algebra for this fixed knot family. In this case, we call generators of such descended algebra as primary Vassiliev invariants, so that any Vassiliev invariant is a polynomial in primary Vassiliev invariants.

One-parameter families: the invariants are ${\mathcal{V}}_{d,i}^{{\mathcal{K}}_n}={\sum }_{j=0}^d{a}_{d,j}^{(i)}{n}^j$ with $a_{d,j}^{(i)}\in \mathbb{Q}$.

The conjecture:

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.

Examples and reasonings provide the intuition that a 1-parametric algebra of polynomial Vassiliev invariants always has only a finite number of generators. Still, we currently lack strict mathematical proof.

For twist knots the paper calls ${\mathcal{V}}_2$ the only primary invariant, which “generates all Vassiliev invariants for the whole family of twist knots”, although the higher invariants are polynomials of higher degree in it; generation is ordinary algebra generation, not generation compatible with the order filtration.

Verified locator

Source: https://arxiv.org/abs/2508.02385v2

• URL: https://arxiv.org/abs/2508.02385v2 (the e-print is a tar archive; member main.tex, 1735 lines, md5 aa0b2aed4fb18cc434585a2221902046):
  • the conjecture in the abstract, l. 165;
  • rational values and the algebra, l. 350–352;
  • primary invariants, l. 356;
  • one-parameter polynomial invariants, l. 1114–1117;
  • the conjecture in Section 4.1, l. 1166;
  • twist knots, l. 1228.
• DOI: 10.1016/j.nuclphysb.2025.117268 (https://doi.org/10.1016/j.nuclphysb.2025.117268; Crossref lists Nuclear Physics B, volume 1022, article 117268, January 2026).