Applications of the L-space satellite formula
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914054047531008 |
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| author | Chen, Daren Zemke, Ian Zhou, Hugo |
| author_facet | Chen, Daren Zemke, Ian Zhou, Hugo |
| contents | We give a formula for the $τ$-invariant of a satellite knot $P(K,n)$ when $P$ is an L-space satellite operator. Our formula holds for general L-space satellite operators $P$ when the companion $K$ satisfies $ε(K)=1$. When $ε(K)$ is $0$ or $-1$, we state a formula which requires some additional assumptions on $P$ or $n$. Our main tool is our algorithm which computes the knot Floer complex of satellite knots constructed using L-space satellite operators, which we developed in a previous paper. Our formula for $τ$ recovers many existing formulas for the behavior of $τ$ under satellite operators, including for cables. We apply our formula to questions about the slice genus of satellite knots, showing, e.g., that if $K$ is a knot with $τ(K)=g_4(K)>0$, then satellites of $K$ by L-space satellite operators have the same property. Another application is a proof that L-space satellite operators satisfy a conjecture of Hedden and Pinzón-Caicedo: If $P$ is an L-space satellite operator which acts as a group homomorphism on the smooth concordance group, then $P$ is either the zero operator, the identity operator, or the orientation reversing operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_20288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Applications of the L-space satellite formula Chen, Daren Zemke, Ian Zhou, Hugo Geometric Topology 57K18, 57K10 We give a formula for the $τ$-invariant of a satellite knot $P(K,n)$ when $P$ is an L-space satellite operator. Our formula holds for general L-space satellite operators $P$ when the companion $K$ satisfies $ε(K)=1$. When $ε(K)$ is $0$ or $-1$, we state a formula which requires some additional assumptions on $P$ or $n$. Our main tool is our algorithm which computes the knot Floer complex of satellite knots constructed using L-space satellite operators, which we developed in a previous paper. Our formula for $τ$ recovers many existing formulas for the behavior of $τ$ under satellite operators, including for cables. We apply our formula to questions about the slice genus of satellite knots, showing, e.g., that if $K$ is a knot with $τ(K)=g_4(K)>0$, then satellites of $K$ by L-space satellite operators have the same property. Another application is a proof that L-space satellite operators satisfy a conjecture of Hedden and Pinzón-Caicedo: If $P$ is an L-space satellite operator which acts as a group homomorphism on the smooth concordance group, then $P$ is either the zero operator, the identity operator, or the orientation reversing operator. |
| title | Applications of the L-space satellite formula |
| topic | Geometric Topology 57K18, 57K10 |
| url | https://arxiv.org/abs/2509.20288 |