Quandle presentations of surface knots in 4-manifolds and bridge numbers
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914567140933632 |
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| author | Zhou, Xiaozhou |
| author_facet | Zhou, Xiaozhou |
| contents | The fundamental quandle is an invariant for distinguishing surface knots, yet computable presentations have traditionally been limited to surfaces embedded in the $4$-sphere. Building on the framework of banded unlink diagrams introduced by Hughes, Kim, and Miller, we give a Wirtinger type presentation of the fundamental quandle of surface links in arbitrary $4$-manifolds. As applications, we extend the work of Sato and Tanaka to show that for any $b \geq 4$ and $m \geq 0$, there exist infinitely many pairwise non-local surface knots with bridge number $b$ in $\mathbb{C}P^2 \#m\overline{\mathbb{C}P^2}$, and we distinguish infinite families of surface knots with isomorphic knot groups, extending results of Tanaka and Taniguchi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14593 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quandle presentations of surface knots in 4-manifolds and bridge numbers Zhou, Xiaozhou Geometric Topology 57K45, 57K12, 57M05 The fundamental quandle is an invariant for distinguishing surface knots, yet computable presentations have traditionally been limited to surfaces embedded in the $4$-sphere. Building on the framework of banded unlink diagrams introduced by Hughes, Kim, and Miller, we give a Wirtinger type presentation of the fundamental quandle of surface links in arbitrary $4$-manifolds. As applications, we extend the work of Sato and Tanaka to show that for any $b \geq 4$ and $m \geq 0$, there exist infinitely many pairwise non-local surface knots with bridge number $b$ in $\mathbb{C}P^2 \#m\overline{\mathbb{C}P^2}$, and we distinguish infinite families of surface knots with isomorphic knot groups, extending results of Tanaka and Taniguchi. |
| title | Quandle presentations of surface knots in 4-manifolds and bridge numbers |
| topic | Geometric Topology 57K45, 57K12, 57M05 |
| url | https://arxiv.org/abs/2605.14593 |