Extending Quotients of Knot Groups over Surfaces in $B^4$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908930173566976 |
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| author | Kjuchukova, Alexandra Orr, Kent E. |
| author_facet | Kjuchukova, Alexandra Orr, Kent E. |
| contents | Let $K\subseteq S^3$ be a knot with exterior $E_K$, and denote by $ρ\colon π_1(E_K)\twoheadrightarrow G$ a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface $F\subseteq B^4$ with $\partial F = K$ over whose exterior $ρ$ extends surjectively. Equivalently, we determine whether the cover of $S^3$ branched over $K$ and induced by $ρ$ bounds a connected cover of $B^4$ branched along such a surface. When $G$ is a dihedral group, we show the obstruction can be computed by evaluating the Seifert form of $K$ on a single curve, a so-called characteristic knot associated to $ρ$. When the dihedral obstruction vanishes, we construct the surface $F$ explicitly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00460 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extending Quotients of Knot Groups over Surfaces in $B^4$ Kjuchukova, Alexandra Orr, Kent E. Geometric Topology Let $K\subseteq S^3$ be a knot with exterior $E_K$, and denote by $ρ\colon π_1(E_K)\twoheadrightarrow G$ a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface $F\subseteq B^4$ with $\partial F = K$ over whose exterior $ρ$ extends surjectively. Equivalently, we determine whether the cover of $S^3$ branched over $K$ and induced by $ρ$ bounds a connected cover of $B^4$ branched along such a surface. When $G$ is a dihedral group, we show the obstruction can be computed by evaluating the Seifert form of $K$ on a single curve, a so-called characteristic knot associated to $ρ$. When the dihedral obstruction vanishes, we construct the surface $F$ explicitly. |
| title | Extending Quotients of Knot Groups over Surfaces in $B^4$ |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2604.00460 |