Local Knots, $ν^+$-Sharp Knots, and Rational Slice Genus
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| Format: | Preprint |
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2026
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| _version_ | 1866918397152854016 |
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| author | Park, Junghwan Wu, Zhongtao Yang, Jingling |
| author_facet | Park, Junghwan Wu, Zhongtao Yang, Jingling |
| contents | Hom and Wu introduced the knot concordance invariant $ν^{+}$ for knots in $S^{3}$ and proved that it gives a lower bound for the slice genus. Wu and Yang extended $ν^{+}$ to knots in rational homology $3$-spheres, where it gives a lower bound for the rational slice genus, an analogue of the slice genus for knots in rational homology $3$-spheres. We call a knot $ν^{+}$-sharp if this bound is realized as an equality.
An open question asks whether a local knot in a $3$-manifold $Y$, that is, a knot contained in a $3$-ball, can bound a surface of smaller genus in $Y\times I$ than in $S^{3}\times I$. Using the Heegaard Floer invariant $ν^+$, we show that this does not occur for local knots arising from $ν^+$-sharp knots: if $K\subset S^3$ is $ν^+$-sharp and $Y$ is a rational homology $3$-sphere, then the induced local knot in $Y$ has rational slice genus equal to the slice genus of $K$. The proof proceeds by establishing an additivity result for the rational slice genus. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18619 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Knots, $ν^+$-Sharp Knots, and Rational Slice Genus Park, Junghwan Wu, Zhongtao Yang, Jingling Geometric Topology 57K10, 57K18, 57R58 Hom and Wu introduced the knot concordance invariant $ν^{+}$ for knots in $S^{3}$ and proved that it gives a lower bound for the slice genus. Wu and Yang extended $ν^{+}$ to knots in rational homology $3$-spheres, where it gives a lower bound for the rational slice genus, an analogue of the slice genus for knots in rational homology $3$-spheres. We call a knot $ν^{+}$-sharp if this bound is realized as an equality. An open question asks whether a local knot in a $3$-manifold $Y$, that is, a knot contained in a $3$-ball, can bound a surface of smaller genus in $Y\times I$ than in $S^{3}\times I$. Using the Heegaard Floer invariant $ν^+$, we show that this does not occur for local knots arising from $ν^+$-sharp knots: if $K\subset S^3$ is $ν^+$-sharp and $Y$ is a rational homology $3$-sphere, then the induced local knot in $Y$ has rational slice genus equal to the slice genus of $K$. The proof proceeds by establishing an additivity result for the rational slice genus. |
| title | Local Knots, $ν^+$-Sharp Knots, and Rational Slice Genus |
| topic | Geometric Topology 57K10, 57K18, 57R58 |
| url | https://arxiv.org/abs/2603.18619 |