The $V_1$- and $V_2$-polynomials of a long virtual knot
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917216998391808 |
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| author | Satoh, Shin Wada, Kodai |
| author_facet | Satoh, Shin Wada, Kodai |
| contents | We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental properties and show that any pair of Laurent polynomials can be realized as $(V_1(K;t),V_2(K;t))$ for some long virtual knot $K$. While these polynomials are not finite type invariants of any degree with respect to virtualizations, their first derivatives at $t=1$ define finite type invariants of degree three. As an application, we obtain an explicit Gauss diagram formula for the $α_3$-invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $V_1$- and $V_2$-polynomials of a long virtual knot Satoh, Shin Wada, Kodai Geometric Topology 57K12, 57K14, 57K16 We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental properties and show that any pair of Laurent polynomials can be realized as $(V_1(K;t),V_2(K;t))$ for some long virtual knot $K$. While these polynomials are not finite type invariants of any degree with respect to virtualizations, their first derivatives at $t=1$ define finite type invariants of degree three. As an application, we obtain an explicit Gauss diagram formula for the $α_3$-invariant. |
| title | The $V_1$- and $V_2$-polynomials of a long virtual knot |
| topic | Geometric Topology 57K12, 57K14, 57K16 |
| url | https://arxiv.org/abs/2601.15634 |