# Give an example of a bounded domain and a piecewise $C^1$ closed curve satisfy given conditions.

1. Give an example of a bounded domain $$\Omega \subset \mathbb {C}$$ and a piecewise $$C^1$$ closed curve $$f$$ in $$\Omega$$ such that $$I(f;z)=5$$ for some $$z \in \mathbb {C}/\Omega$$. (Here $$C^1$$ means the components have continuous derivatives for all t within the interval $$[a,b]$$. And $$I(f;z)=5$$ is the winding number of $$f$$ on $$z$$.)

2. Give an example of a bounded domain $$\Omega \subset \mathbb {C}$$ and a cycle $$\Gamma=\rho_1 + \rho_2 + \dots +\rho_s$$ for some $$s\in \mathbb {Z}_+$$, such that

• each $$\rho_i$$ is a $$C^1$$ simple closed cuve in $$\Omega$$,
• no two $$\rho_i, \rho_j$$ intersect, and
• for every $$k \in {1, \dots,5}$$ there is a point $$a_k\in {C}/\Omega$$ such that $$I(\Gamma;a_k)=k$$.