Equivariant algebraic concordance of strongly invertible knots
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912065373863936 |
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| author | Di Prisa, Alessio |
| author_facet | Di Prisa, Alessio |
| contents | By considering a particular type of invariant Seifert surfaces we define a homomorphism $Φ$ from the (topological) equivariant concordance group of directed strongly invertible knots $\widetilde{\mathcal{C}}$ to a new equivariant algebraic concordance group $\widetilde{\mathcal{G}}^{\mathbb{Z}}$. We prove that $Φ$ lifts both Miller and Powell's equivariant algebraic concordance homomorphism, and Alfieri and Boyle's equivariant signature. Moreover, we provide a partial result on the isomorphism type of $\widetilde{\mathcal{G}}^{\mathbb{Z}}$, and we obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox-Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that $Φ$ can obstruct equivariant sliceness for knots with Alexander polynomial one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_11895 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equivariant algebraic concordance of strongly invertible knots Di Prisa, Alessio Geometric Topology 57K10 (Primary), 57M60, 57R85 (Secondary) By considering a particular type of invariant Seifert surfaces we define a homomorphism $Φ$ from the (topological) equivariant concordance group of directed strongly invertible knots $\widetilde{\mathcal{C}}$ to a new equivariant algebraic concordance group $\widetilde{\mathcal{G}}^{\mathbb{Z}}$. We prove that $Φ$ lifts both Miller and Powell's equivariant algebraic concordance homomorphism, and Alfieri and Boyle's equivariant signature. Moreover, we provide a partial result on the isomorphism type of $\widetilde{\mathcal{G}}^{\mathbb{Z}}$, and we obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox-Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that $Φ$ can obstruct equivariant sliceness for knots with Alexander polynomial one. |
| title | Equivariant algebraic concordance of strongly invertible knots |
| topic | Geometric Topology 57K10 (Primary), 57M60, 57R85 (Secondary) |
| url | https://arxiv.org/abs/2303.11895 |