Iterated satellite operators on the knot concordance group
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929739808112640 |
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| author | Cha, Jae Choon Kim, Taehee |
| author_facet | Cha, Jae Choon Kim, Taehee |
| contents | We show that for a winding number zero satellite operator $P$ on the knot concordance group, if the axis of $P$ has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration $P^n$ generates an infinite rank subgroup for each $n$. Furthermore, the graded quotients of the filtration of the knot concordance group associated with $P$ have infinite rank at all levels. This gives an affirmative answer to a question of Hedden and Pinzón-Caicedo in many cases. We also show that under the same hypotheses, $P^n$ is not a homomorphism on the knot concordance group for each $n$. We use amenable $L^2$-signatures to prove these results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_04629 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Iterated satellite operators on the knot concordance group Cha, Jae Choon Kim, Taehee Geometric Topology 57K10, 57N70 We show that for a winding number zero satellite operator $P$ on the knot concordance group, if the axis of $P$ has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration $P^n$ generates an infinite rank subgroup for each $n$. Furthermore, the graded quotients of the filtration of the knot concordance group associated with $P$ have infinite rank at all levels. This gives an affirmative answer to a question of Hedden and Pinzón-Caicedo in many cases. We also show that under the same hypotheses, $P^n$ is not a homomorphism on the knot concordance group for each $n$. We use amenable $L^2$-signatures to prove these results. |
| title | Iterated satellite operators on the knot concordance group |
| topic | Geometric Topology 57K10, 57N70 |
| url | https://arxiv.org/abs/2402.04629 |