A Seifert algorithm for integral homology spheres
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914808177098752 |
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| author | Alegria, Linda V. Menasco, William W. |
| author_facet | Alegria, Linda V. Menasco, William W. |
| contents | From classical knot theory we know that every knot in $S^3$ is the boundary of an oriented, embedded surface. A standard demonstration of this fact achieved by elementary technique comes from taking a regular projection of any knot and employing Seifert's constructive algorithm. In this note we give a natural generalization of Seifert's algorithm to any closed integral homology 3-sphere. The starting point of our algorithm is presenting the handle structure of a Heegaard splitting of a given integral homology sphere as a planar diagram on the boundary of a $3$-ball. (For a well known example of such a planar presentation, see the Poincaré homology sphere planar presentation in {\em Knots and Links} by D. Rolfsen \cite{Rolfsen}.) An oriented link can then be represented by the regular projection of an oriented $k$-strand tangle. From there we give a natural way to find a ``Seifert circle" and associated half-twisted bands. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Seifert algorithm for integral homology spheres Alegria, Linda V. Menasco, William W. Geometric Topology 57K10 (Primary), 57K30 (Secondary) From classical knot theory we know that every knot in $S^3$ is the boundary of an oriented, embedded surface. A standard demonstration of this fact achieved by elementary technique comes from taking a regular projection of any knot and employing Seifert's constructive algorithm. In this note we give a natural generalization of Seifert's algorithm to any closed integral homology 3-sphere. The starting point of our algorithm is presenting the handle structure of a Heegaard splitting of a given integral homology sphere as a planar diagram on the boundary of a $3$-ball. (For a well known example of such a planar presentation, see the Poincaré homology sphere planar presentation in {\em Knots and Links} by D. Rolfsen \cite{Rolfsen}.) An oriented link can then be represented by the regular projection of an oriented $k$-strand tangle. From there we give a natural way to find a ``Seifert circle" and associated half-twisted bands. |
| title | A Seifert algorithm for integral homology spheres |
| topic | Geometric Topology 57K10 (Primary), 57K30 (Secondary) |
| url | https://arxiv.org/abs/2405.14805 |