Volume 32, Issue 4
Subsurface 1-Distance of the Handlebody

Commun. Math. Res., 32 (2016), pp. 375-382.

Published online: 2021-05

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• Abstract

For a handlebody $H$ with $∂H = S$, let $F ⊂ S$ be an essential connected subsurface of $S$. Let $\mathcal{C}(S)$ be the curve complex of $S$, $\mathcal{AC}(F)$ be the arc and curve complex of $F$, $\mathcal{D}(H) ⊂ \mathcal{C}(S)$ be the disk complex of $H$ and $π_F (\mathcal{D}(H)) ⊂ \mathcal{AC}(F)$ be the image of $\mathcal{D}(H)$ in $\mathcal{AC}(F)$. We introduce the definition of subsurface 1-distance between the 1-simplices of $\mathcal{AC}(F)$ and show that under some hypothesis, $π_F (\mathcal{D}(H))$ comes within subsurface 1-distance at most 4 of every 1-simplex of $\mathcal{AC}(F)$.

• Keywords

handlebody, curve complex, arc and curve complex, subsurface 1-distance.

57M99

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@Article{CMR-32-375, author = {Dongqi and Sun and and 18587 and and Dongqi Sun}, title = {Subsurface 1-Distance of the Handlebody}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {32}, number = {4}, pages = {375--382}, abstract = {

For a handlebody $H$ with $∂H = S$, let $F ⊂ S$ be an essential connected subsurface of $S$. Let $\mathcal{C}(S)$ be the curve complex of $S$, $\mathcal{AC}(F)$ be the arc and curve complex of $F$, $\mathcal{D}(H) ⊂ \mathcal{C}(S)$ be the disk complex of $H$ and $π_F (\mathcal{D}(H)) ⊂ \mathcal{AC}(F)$ be the image of $\mathcal{D}(H)$ in $\mathcal{AC}(F)$. We introduce the definition of subsurface 1-distance between the 1-simplices of $\mathcal{AC}(F)$ and show that under some hypothesis, $π_F (\mathcal{D}(H))$ comes within subsurface 1-distance at most 4 of every 1-simplex of $\mathcal{AC}(F)$.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2016.04.09}, url = {http://global-sci.org/intro/article_detail/cmr/18909.html} }
TY - JOUR T1 - Subsurface 1-Distance of the Handlebody AU - Sun , Dongqi JO - Communications in Mathematical Research VL - 4 SP - 375 EP - 382 PY - 2021 DA - 2021/05 SN - 32 DO - http://doi.org/10.13447/j.1674-5647.2016.04.09 UR - https://global-sci.org/intro/article_detail/cmr/18909.html KW - handlebody, curve complex, arc and curve complex, subsurface 1-distance. AB -

For a handlebody $H$ with $∂H = S$, let $F ⊂ S$ be an essential connected subsurface of $S$. Let $\mathcal{C}(S)$ be the curve complex of $S$, $\mathcal{AC}(F)$ be the arc and curve complex of $F$, $\mathcal{D}(H) ⊂ \mathcal{C}(S)$ be the disk complex of $H$ and $π_F (\mathcal{D}(H)) ⊂ \mathcal{AC}(F)$ be the image of $\mathcal{D}(H)$ in $\mathcal{AC}(F)$. We introduce the definition of subsurface 1-distance between the 1-simplices of $\mathcal{AC}(F)$ and show that under some hypothesis, $π_F (\mathcal{D}(H))$ comes within subsurface 1-distance at most 4 of every 1-simplex of $\mathcal{AC}(F)$.

Dongqi Sun. (2021). Subsurface 1-Distance of the Handlebody. Communications in Mathematical Research . 32 (4). 375-382. doi:10.13447/j.1674-5647.2016.04.09
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