An unoriented analogue of slice-torus invariant
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911837161783296 |
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| author | Sato, Kouki |
| author_facet | Sato, Kouki |
| contents | A slice-torus invariant is an $\mathbb{R}$-valued homomorphism on the knot concordance group whose value gives a lower bound for the 4-genus such that the equality holds for any positive torus knot. Such invariants have been discovered in many of knot homology theories, while it is known that any slice-torus invariant does not factor through the topological concordance group. In this paper, we introduce the notion of "unoriented slice-torus invariant", which can be regarded as the same as slice-torus invariant except for the condition about the orientability of surfaces. Then we show that the Ozsváth-Stipsicz-Szabó $\upsilon$-invariant, the Ballinger $t$-invariant and the Daemi-Scaduto $h_{\mathbb{Z}}$-invariant (shifted by a half of the knot signature) are unoriented slice-torus invariants. As an application, we give a new method for computing the above invariants, which is analogous to Livingston's method for computing slice-torus invariants. Moreover, we use the method to prove that any unoriented slice-torus invariant does not factor through the topological concordance group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_04056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An unoriented analogue of slice-torus invariant Sato, Kouki Geometric Topology 57K18, 57K10 A slice-torus invariant is an $\mathbb{R}$-valued homomorphism on the knot concordance group whose value gives a lower bound for the 4-genus such that the equality holds for any positive torus knot. Such invariants have been discovered in many of knot homology theories, while it is known that any slice-torus invariant does not factor through the topological concordance group. In this paper, we introduce the notion of "unoriented slice-torus invariant", which can be regarded as the same as slice-torus invariant except for the condition about the orientability of surfaces. Then we show that the Ozsváth-Stipsicz-Szabó $\upsilon$-invariant, the Ballinger $t$-invariant and the Daemi-Scaduto $h_{\mathbb{Z}}$-invariant (shifted by a half of the knot signature) are unoriented slice-torus invariants. As an application, we give a new method for computing the above invariants, which is analogous to Livingston's method for computing slice-torus invariants. Moreover, we use the method to prove that any unoriented slice-torus invariant does not factor through the topological concordance group. |
| title | An unoriented analogue of slice-torus invariant |
| topic | Geometric Topology 57K18, 57K10 |
| url | https://arxiv.org/abs/2404.04056 |