Suppose I have a smooth orientable surface $Q$ and a compact region $R$ of $Q$. Suppose there is a closed curve $C$ that divides R into two connected components $R_1,R_2$ but does not divide Q into separate regions. Suppose there is a (smooth) curve $f$ with one endpoint in $R_1$, the other in $R_2$ and is otherwise disjoint from $R$. Must there it be possible to find a closed curve in Q not intersecting R, that does not divide Q into two separate regions? Colloquially must it be possible to find a handle we can "cut" from our surface?
Figure below: Region $R$ in orange is a subset of our surface which is the torus. The curve $C$ in blue divides $R$ into $R_1$ and $R_2$. The red curve has as two endpoints on the boundaries of $R_1$ and $R_2$ respectively. The green curve, disjoint from $R$ does not divide $Q$\ $R$ into two disjoint regions.
Also anyone recommend a basic reference for learning how to prove this kind of thing?(smooth 2dimensional surfaces only focusing on genus and surface operations) Basic topology by Armstrong is understandable but doesn't provide enough tools for this kind of thing. Other textbooks go too far and go beyond 2 dimensions or are a bit too abstract.