Taut foliations, braid positivity, and unknot detection
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915226359693312 |
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| author | Krishna, Siddhi |
| author_facet | Krishna, Siddhi |
| contents | We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r < 2g(K)-1$, the 3-manifold obtained via $r$-framed Dehn surgery along $K$ admits a taut foliation. Our main result provides some positive evidence towards this conjecture: we construct taut foliations in such manifolds whenever $r<g(K)+1$. As an application, we produce a novel braid positivity obstruction for cable knots by proving that the $(n,\pm 1)$-cable of a knot $K$ is braid positive if and only if $K$ is the unknot. We also present some curious examples demonstrating the limitations of our construction; these examples can also be viewed as providing some negative evidence towards the L-space conjecture. Finally, we apply our main result to produce taut foliations in some splicings of knot exteriors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Taut foliations, braid positivity, and unknot detection Krishna, Siddhi Geometric Topology We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r < 2g(K)-1$, the 3-manifold obtained via $r$-framed Dehn surgery along $K$ admits a taut foliation. Our main result provides some positive evidence towards this conjecture: we construct taut foliations in such manifolds whenever $r<g(K)+1$. As an application, we produce a novel braid positivity obstruction for cable knots by proving that the $(n,\pm 1)$-cable of a knot $K$ is braid positive if and only if $K$ is the unknot. We also present some curious examples demonstrating the limitations of our construction; these examples can also be viewed as providing some negative evidence towards the L-space conjecture. Finally, we apply our main result to produce taut foliations in some splicings of knot exteriors. |
| title | Taut foliations, braid positivity, and unknot detection |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2312.00196 |