Work

Algebraic structures of Vassiliev invariants for knot families

Year: 2025

Type: article

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 ≤ 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.

Source: Nuclear Physics B

DOI: 10.1016/j.nuclphysb.2025.117268

Authors: E. Lanina, A. Sleptsov

Institutions: Moscow Institute of Physics and Technology, Kurchatov Institute, Institute for Information Transmission Problems

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Ministry of Science and Higher Education of the Russian Federation, Moscow Institute of Physics and Technology

Award: G2622452862