  Let $\kappa\in(4,8)$. Let $\gamma$ be an SLE$_\kappa$ curve in a Jordan
domain $D$ connecting $a_1\ne a_2\in\partial D$. We attach $\gamma$ with two
open boundary arcs $A_1,A_2$ of $D$, which share end points $b_1\ne
b_2\in\partial D\setminus\{a_1,a_2\}$, and consider for each $z_0\in D$ the
limit $$ \lim_{r \downarrow 0}r^{1-\frac 38\kappa} \mathbb{P}[\gamma\cup
A_1\cup A_2 \mbox{ has a cut point in }\{|z-z_0|<r\}].$$ We prove that the
limit converges, derive a rate of convergence, and obtain the exact formula of
the limit up to a multiplicative constant depending only on $\kappa$.
