of $S_g$ isometrically imbedded in $\mathbb{R}^3$. In particular, ${\operatorname{{\mathbb S}}}_g(1) \cong S_g$, and hence we shall write $S_g$ for ${\operatorname{{\mathbb S}}}_g(1)$.

(iv) Given two surfaces $S_{g_1}$ and ${\operatorname{{\mathbb S}}}_{g_2}(m)$ and a fixed $k\in \mathbb{N}$, we construct a new surface $S_g$ with $g=(g_1+mg_2 + (k-1)m)$, containing a multicurve of type ${\operatorname{{\mathcal{C}}}}^{(k)}(m)$, in the following manner. We remove $km$ disks $\{D_{i,j}^1 : 1\leq j\leq k, 1\leq i\leq m\}$ on $S_{g_1}$ and $k$ disks $\{D_{i,j}^2 : 1\leq j\leq k\}$ on each $S_{g_2}^i$. Now connect $\partial D_{i,j}^1$ to $\partial D_{i,j}^2$ along a 1-handle $A_{i,j}$, and choose the unique curve (upto isotopy) $c_{i,j}$ on $A_{i,j}$. Let ${\operatorname{{\mathcal{C}}}}= \{c_{i,j}\}$, then note that ${\operatorname{{\mathcal{C}}}}= {\operatorname{{\mathcal{C}}}}^{(k)}(m)$, so we write $$S_{g_1}\#_{{\operatorname{{\mathcal{C}}}}} {\operatorname{{\mathbb S}}}_{g_2}(m)$$ for the new surface $S_g$.

(v) Similarly, given surfaces $\{S_{g_1}, {\operatorname{{\mathbb S}}}_{g_2,1}(m_1), \ldots, {\operatorname{{\mathbb S}}}_{g_2,s}(m_s)\}$ and non-negative integers $\{k_1,k_2,\ldots, k_s\}$, we construct a new surface $S_g$ with $g= g_1 + \sum_{i=1}^s m_i(g_{2,i} + k_i-1)$, containing a multicurve of type ${\operatorname{{\mathcal{C}}}}= \sqcup_{i=1}^s {\operatorname{{\mathcal{C}}}}^{(k_i)}(m_i)$. Let $$S_{g_i'} := S_{g_1}\#_{{\operatorname{{\mathcal{C}}}}^{(k_i)}(m_i)} {\operatorname{{\mathbb S}}}_{g_2,i}(m_i)$$ and ${\operatorname{{\mathcal{C}}}}_i := {\operatorname{{\mathcal{C}}}}\setminus {\operatorname{{\mathcal{C}}}}^{(k_i)}(m_i)$, we now define $${\overline{\textbf{\Large \#}}}_{i=1}^s \left( S_{g_1}{\#_{{\operatorname{{\mathcal{C}}}}^{(k_i)}(m_i)}} {\operatorname{{\mathbb S}}}_{g_2,i}(m_i) \right) := \bigcup_{i=1}^s \widehat{S_{g_i'}({\operatorname{{\mathcal{C}}}}_i)}.$$ If $s = 2$, we simply write $S_g = {\operatorname{{\mathbb S}}}_{g_{2,1}}(m_1)\#_{{\operatorname{{\mathcal{C}}}}^{(k_1)}(m_1)} S_{g_1}\#_{{\operatorname{{\mathcal{C}}}}^{(k_2)}(m_2)} {\operatorname{{\mathbb S}}}_{g_{2,2}}(m_2)$.

In Figure \[fig:sg\_sgm\] below, we give 