The intersection polynomials of a long virtual knot I: Definitions and properties
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908694822780928 |
|---|---|
| author | Nakamura, Takuji Nakanishi, Yasutaka Satoh, Shin Wada, Kodai |
| author_facet | Nakamura, Takuji Nakanishi, Yasutaka Satoh, Shin Wada, Kodai |
| contents | We introduce twelve polynomial invariants for long virtual knots, called intersection polynomials, extending and refining the three intersection polynomials for virtual knots. They are defined via intersection numbers of cycles on a closed surface, considering the order of over- and under-crossings. We study their fundamental properties including behavior under symmetries, crossing changes, and concatenation products. All are finite-type invariants of degree two under crossing changes, but not under virtualizations, and we examine their relation to the closure and the values at $t=1$ of their derivatives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05366 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The intersection polynomials of a long virtual knot I: Definitions and properties Nakamura, Takuji Nakanishi, Yasutaka Satoh, Shin Wada, Kodai Geometric Topology Primary 57K12, Secondary 57K14, 57K16 We introduce twelve polynomial invariants for long virtual knots, called intersection polynomials, extending and refining the three intersection polynomials for virtual knots. They are defined via intersection numbers of cycles on a closed surface, considering the order of over- and under-crossings. We study their fundamental properties including behavior under symmetries, crossing changes, and concatenation products. All are finite-type invariants of degree two under crossing changes, but not under virtualizations, and we examine their relation to the closure and the values at $t=1$ of their derivatives. |
| title | The intersection polynomials of a long virtual knot I: Definitions and properties |
| topic | Geometric Topology Primary 57K12, Secondary 57K14, 57K16 |
| url | https://arxiv.org/abs/2512.05366 |