On Legendrian representatives of non-fibered knots
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909371121795072 |
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| author | Li, Zhenkun Wan, Shunyu |
| author_facet | Li, Zhenkun Wan, Shunyu |
| contents | We show that in $(S^3,ξ_{std})$ if $K$ is a non-trivial knot that realizes the three-dimensional Thurston-Bennequin bound (i.e. $K$ has a Legendrian representative $Λ$ with $tb(Λ)-rot(Λ)=2g(K)-1$), then $K$ has a Legendrian representative $L$ with $tb=0$. Moreover, this result can be easily generalized to contact manifolds that uniquely represent the associated contact invariants. This is the first result on Legendrian representatives of non-fibered knots in $3-$manifolds other than $S^3$. We also show that if $K$ is a nearly fibered knot in $S^3$ then $τ(K)=g(K)$ implies that $K$ realizes the three-dimensional Thurston-Bennequin bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16549 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Legendrian representatives of non-fibered knots Li, Zhenkun Wan, Shunyu Geometric Topology We show that in $(S^3,ξ_{std})$ if $K$ is a non-trivial knot that realizes the three-dimensional Thurston-Bennequin bound (i.e. $K$ has a Legendrian representative $Λ$ with $tb(Λ)-rot(Λ)=2g(K)-1$), then $K$ has a Legendrian representative $L$ with $tb=0$. Moreover, this result can be easily generalized to contact manifolds that uniquely represent the associated contact invariants. This is the first result on Legendrian representatives of non-fibered knots in $3-$manifolds other than $S^3$. We also show that if $K$ is a nearly fibered knot in $S^3$ then $τ(K)=g(K)$ implies that $K$ realizes the three-dimensional Thurston-Bennequin bound. |
| title | On Legendrian representatives of non-fibered knots |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2405.16549 |