Let $\gamma :[t_0,t_1]\to \mathbb{C}$ be a continuous curve. We say a disk chain $((C_0,c_0),\dots ,(C_n,c_n))$ goes along $\gamma$ if $c_k=\gamma ({\tau }_k)$ for ${\tau }_k\in [t_0,t_1]$, and $t_0={\tau }_0<{\tau }_1<\cdots <{\tau }_n:=t_1$ and each for each segment $[{\tau }_{k-1},{\tau }_k]$, we have that1;; $\gamma ([{\tau }_{k-1},{\tau }_k])\subseteq C_{k-1}\bigcap C_k$.