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 - <p style="text-align: justify;">For a handlebody $H$ with $∂H = S$, let $F ⊂ S$ be an essential connected
subsurface of $S$. Let&nbsp;$\mathcal{C}(S)$ be the curve complex of $S$,&nbsp;$\mathcal{AC}(F)$ be the arc and curve
complex of $F$,&nbsp;$\mathcal{D}(H) ⊂&nbsp;\mathcal{C}(S)$ be the disk complex of $H$ and $π_F (\mathcal{D}(H)) ⊂&nbsp;\mathcal{AC}(F)$ be
the image of&nbsp;$\mathcal{D}(H)$ in&nbsp;$\mathcal{AC}(F)$. 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 $\mathcal{AC}(F)$.</p>