The band connected sum and the second Kirby move for higher-dimensional links
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911351112204288 |
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| author | Skopenkov, A. |
| author_facet | Skopenkov, A. |
| contents | Let $f:S^q\sqcup S^q\to S^m$ be a link (i.e. an embedding).
How does (the isotopy class of) the knot $S^q\to S^m$ obtained by embedded connected sum of the components of $f$ depend on $f$?
Define a link $σf:S^q\sqcup S^q\to S^m$ as follows. The first component of $σf$ is the `standardly shifted' first component of $f$. The second component of $σf$ is the embedded connected sum of the components of $f$. How does (the isotopy class of) $σf$ depend on $f$?
How does (the isotopy class of) the link $S^q\sqcup S^q\to S^m$ obtained by embedded connected sum of the last two components of a link $g:S^q_1\sqcup S^q_2\sqcup S^q_3\to S^m$ depend on $g$?
We give the answers for the `first non-trivial case' $q=4k-1$ and $m=6k$. The first answer was used by S. Avvakumov for classification of linked 3-manifolds in $S^6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The band connected sum and the second Kirby move for higher-dimensional links Skopenkov, A. Geometric Topology Algebraic Topology 57R40, 55Q25 Let $f:S^q\sqcup S^q\to S^m$ be a link (i.e. an embedding). How does (the isotopy class of) the knot $S^q\to S^m$ obtained by embedded connected sum of the components of $f$ depend on $f$? Define a link $σf:S^q\sqcup S^q\to S^m$ as follows. The first component of $σf$ is the `standardly shifted' first component of $f$. The second component of $σf$ is the embedded connected sum of the components of $f$. How does (the isotopy class of) $σf$ depend on $f$? How does (the isotopy class of) the link $S^q\sqcup S^q\to S^m$ obtained by embedded connected sum of the last two components of a link $g:S^q_1\sqcup S^q_2\sqcup S^q_3\to S^m$ depend on $g$? We give the answers for the `first non-trivial case' $q=4k-1$ and $m=6k$. The first answer was used by S. Avvakumov for classification of linked 3-manifolds in $S^6$. |
| title | The band connected sum and the second Kirby move for higher-dimensional links |
| topic | Geometric Topology Algebraic Topology 57R40, 55Q25 |
| url | https://arxiv.org/abs/2406.15367 |