On the multiplicity of Knot Floer order under cabling
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909805598212096 |
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| author | Suchodoll, David |
| author_facet | Suchodoll, David |
| contents | The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq \operatorname{br}(K)$. For all $(p,q)$-cables of L-space knots, we show that $\operatorname{Ord}(K) + 1$ is multiplicative in $p$ when $g(K) > 1$, and the same holds for $g(K) = 1$ provided $q > 2p$. We also compute the knot Floer order in the range $q < 2p$, thereby determining $\operatorname{Ord}(K_{p,q})$ in terms of $\operatorname{Ord}(K)$ for all cables of L-space knots. We establish upper bounds under cabling for $\operatorname{Ord}(K)$ and discuss potential applications to a conjecture by Krishna and Morton, proving that the braid index of an L-space cable appears as an exponent in its Alexander polynomial if it does for its companion, provided $\operatorname{Ord}(K)+1$ is multiplicative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the multiplicity of Knot Floer order under cabling Suchodoll, David Geometric Topology 57K10, 57K14, 57K16, 57K18 The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq \operatorname{br}(K)$. For all $(p,q)$-cables of L-space knots, we show that $\operatorname{Ord}(K) + 1$ is multiplicative in $p$ when $g(K) > 1$, and the same holds for $g(K) = 1$ provided $q > 2p$. We also compute the knot Floer order in the range $q < 2p$, thereby determining $\operatorname{Ord}(K_{p,q})$ in terms of $\operatorname{Ord}(K)$ for all cables of L-space knots. We establish upper bounds under cabling for $\operatorname{Ord}(K)$ and discuss potential applications to a conjecture by Krishna and Morton, proving that the braid index of an L-space cable appears as an exponent in its Alexander polynomial if it does for its companion, provided $\operatorname{Ord}(K)+1$ is multiplicative. |
| title | On the multiplicity of Knot Floer order under cabling |
| topic | Geometric Topology 57K10, 57K14, 57K16, 57K18 |
| url | https://arxiv.org/abs/2506.09577 |