Instanton dimensions of knot surgeries over arbitrary fields
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| Format: | Preprint |
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2025
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| _version_ | 1866914167266476032 |
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| author | Li, Zhenkun Ye, Fan |
| author_facet | Li, Zhenkun Ye, Fan |
| contents | Suppose $K \subset S^3$ is a knot and suppose $p$ and $q$ are co-prime integers with $q\ge 1$. For any field $\mathbb{K}$, we establish a dimension formula for the framed instanton homology of knot surgeries: $$ \dim I^\sharp(S^3_{p/q}(K); \mathbb{K}) = q \cdot r_{\mathbb{K}}(K) + |p - q \cdot ν^\sharp_{\mathbb{K}}(K)| $$ for certain integers $r_{\mathbb{K}}(K)$ and $ν^\sharp_{\mathbb{K}}(K)$, except possibly when $p/q = ν^\sharp_{\mathbb{K}}(K)$ and $ν^\sharp_{\mathbb{K}}(K)$ is even. This formula generalizes the result of Baldwin--Sivek from the case $\mathbb{K} = \mathbb{C}$ to arbitrary fields. Based on the result for $\mathbb{K} = \mathbb{Z}/2$, we obtain that $S^3_{p/q}(K)$ is not $SU(2)$-abelian for any knot $K$ other than the unknot and the right-handed trefoil whenever $p/q \in [0,6)$ and $p \in \{ a^e, 2a^e \}$ for some prime number $a$ and natural number $e$, thereby extending existing results for $p/q \in [0,5]$ and $p = a^e$. A byproduct of the techniques developed in this paper is that we generalize the distance-two surgery exact triangle by Culler--Daemi--Xie and Daemi--Miller-Eismeier--Lidman from $\mathbb{Z}/2$ coefficients to any coefficient ring. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_17877 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Instanton dimensions of knot surgeries over arbitrary fields Li, Zhenkun Ye, Fan Geometric Topology Suppose $K \subset S^3$ is a knot and suppose $p$ and $q$ are co-prime integers with $q\ge 1$. For any field $\mathbb{K}$, we establish a dimension formula for the framed instanton homology of knot surgeries: $$ \dim I^\sharp(S^3_{p/q}(K); \mathbb{K}) = q \cdot r_{\mathbb{K}}(K) + |p - q \cdot ν^\sharp_{\mathbb{K}}(K)| $$ for certain integers $r_{\mathbb{K}}(K)$ and $ν^\sharp_{\mathbb{K}}(K)$, except possibly when $p/q = ν^\sharp_{\mathbb{K}}(K)$ and $ν^\sharp_{\mathbb{K}}(K)$ is even. This formula generalizes the result of Baldwin--Sivek from the case $\mathbb{K} = \mathbb{C}$ to arbitrary fields. Based on the result for $\mathbb{K} = \mathbb{Z}/2$, we obtain that $S^3_{p/q}(K)$ is not $SU(2)$-abelian for any knot $K$ other than the unknot and the right-handed trefoil whenever $p/q \in [0,6)$ and $p \in \{ a^e, 2a^e \}$ for some prime number $a$ and natural number $e$, thereby extending existing results for $p/q \in [0,5]$ and $p = a^e$. A byproduct of the techniques developed in this paper is that we generalize the distance-two surgery exact triangle by Culler--Daemi--Xie and Daemi--Miller-Eismeier--Lidman from $\mathbb{Z}/2$ coefficients to any coefficient ring. |
| title | Instanton dimensions of knot surgeries over arbitrary fields |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2511.17877 |