For those who may feel they need the definitions in prime end theory, I provided what I think is sufficient below. A lot of what's below comes from a paper by John C. Mayer, "Inequivalent Embeddings and Prime Ends," with some details changed. If you are familiar with prime end theory, you can skip reading my definitions and hopefully answer the question that comes after.
Let $X$ be a continuum contained in the 2-sphere, $S^2$. A $\textit{crosscut}$ is an open arc in $S^2 \backslash X$ whose endpoints are contained in $X$. A sequence of crosscuts $( C_i)_{i=1}^\infty$ is a $\textit{chain}$ if $C_i$ converge to a point, no two crosscuts $C_i, C_j$ have a common endpoint, and $C_i$ separates $C_{i-1}$ and $C_{i+1}$ in $S^2 \backslash X$ for each $i = 1, 2, \ldots$. Let $\Delta(C_i)$ denote the simply connected domain in $S^2 \backslash X$ whose boundary contains $C_i$ and so that if $i, j \in \mathbb{N}$ are such that $i < j$, then $\Delta(C_i) \supset \Delta(C_j)$.
A $\textit{prime end}$ of $S^2 \backslash X$ is an equivalence class of chains of crosscuts, in that two chains of cross cuts $(C_i)$ and $(K_i)$ are equivalent if for every $i \in \mathbb{N}$, there is a $j \in \mathbb{N}$ such that $\Delta(C_i) \supset \Delta(K_j)$ and similarly, for every $k \in \mathbb{N}$, there is a $m \in \mathbb{N}$ such that $\Delta(K_k) \supset \Delta(C_m)$.
Given a prime end $E$, the set of principle points of $E$, denoted $P(E)$, is defined to be the set of all points in $X$ for which some chain of crosscuts in $E$ converges. The impression of $E$, denoted $I(E)$, is given by $\bigcap_{i=1}^\infty \overline{\Delta(C_i)}$, where $(C_i)$ is a chain of crosscuts belonging to $E$. In general, $P(E) \subset I(E)$.
A prime end $E$ is called $\textit{type 1}$ if $P(E) = I(E)$, where $P(E)$ and $I(E)$ are degenerate. We say $E$ is $\textit{type 2}$ if $P(E) \neq I(E)$ where only $P(E)$ is degenerate. We say $E$ is $\textit{type 3}$ if $P(E) = I(E)$, both of which are nondegenerate. We say $E$ is $\textit{type 4}$ if $P(E) \neq I(E)$, both of which are nondegenerate.
Now with the supporting definitions, my question is this: What are some examples of planar continua $X$ so that, when embedded in $S^2$, are such that there are prime ends of $S^2\backslash X$ which are type 4? Can someone lead me to literature with such examples? Ones with pictures would be helpful. Or, could someone construct examples which can be drawn according to some procedure? Any help with this would be greatly appreciated.