Algebraic structures of Vassiliev invariants for knot families
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915748706779136 |
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| author | Lanina, E. Sleptsov, A. |
| author_facet | Lanina, E. Sleptsov, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic structures of Vassiliev invariants for knot families Lanina, E. Sleptsov, A. High Energy Physics - Theory Mathematical Physics Geometric Topology Quantum Algebra 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. |
| title | Algebraic structures of Vassiliev invariants for knot families |
| topic | High Energy Physics - Theory Mathematical Physics Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2508.02385 |