Closures of 1-tangles and annulus twists
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909647372288000 |
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| author | Taylor, Scott A. |
| author_facet | Taylor, Scott A. |
| contents | A 1-tangle is a properly embedded arc $ψ$ in an unknotted solid torus $V$ in $S^3$. Attaching an arc $ϕ$ in the complementary solid torus $W$ to its endpoints creates a knot $K(ϕ)$ called the closure of $ψ$. We show that for a given nontrivial 1-tangle $ψ$ there exist at most two closures that are the unknot. We give a general method for producing nontrivial 1-tangles admitting two distinct closures and show that our construction accounts for all such examples. As an application, we show that if we twist an unknot $q \neq 0$ times around an unknotted sufficiently incompressible annulus intersecting it exactly once, then there is at most one $q$ such that the resulting knot is unknotted and, if there is such, then $q = \pm 1$. With additional work, we also show that the Krebes 1-tangle does not admit an unknot closure. Our key tools are the ``wrapping index'' which compares how two complementary 1-tangles $ϕ_1$ and $ϕ_2$ wrap around $W$, a theorem of the author's from sutured manifold theory, and theorems of Gabai and Scharlemann concerning band sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Closures of 1-tangles and annulus twists Taylor, Scott A. Geometric Topology 57K10 A 1-tangle is a properly embedded arc $ψ$ in an unknotted solid torus $V$ in $S^3$. Attaching an arc $ϕ$ in the complementary solid torus $W$ to its endpoints creates a knot $K(ϕ)$ called the closure of $ψ$. We show that for a given nontrivial 1-tangle $ψ$ there exist at most two closures that are the unknot. We give a general method for producing nontrivial 1-tangles admitting two distinct closures and show that our construction accounts for all such examples. As an application, we show that if we twist an unknot $q \neq 0$ times around an unknotted sufficiently incompressible annulus intersecting it exactly once, then there is at most one $q$ such that the resulting knot is unknotted and, if there is such, then $q = \pm 1$. With additional work, we also show that the Krebes 1-tangle does not admit an unknot closure. Our key tools are the ``wrapping index'' which compares how two complementary 1-tangles $ϕ_1$ and $ϕ_2$ wrap around $W$, a theorem of the author's from sutured manifold theory, and theorems of Gabai and Scharlemann concerning band sums. |
| title | Closures of 1-tangles and annulus twists |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2506.11226 |