Suppose $L=L_1 \cup L_2 \cup L_3$ be a classical link of three components. Suppose $L$ is an unlink, that is $L$ can be splitted into three simple closed curves. Assume that $L$ has a diagram in 2-plane such that
- There are some crossings between $L_1$ and $L_2$ And between $L_1$ and $L_3$.
- There are some crossings between $L_2$ and $L_3$.
- There are some self intersections of each $L_i, i=1,2,3$.
I have two questions:
Q1: Can we apply a finite sequence of Reidemeister moves to the self crossings of $L_1$ only so that all self crossings of $L_1$ are eliminated without affecting the other crossings of $L$. If the answer is yes, then this can be generalized to $L_2$ and $L_3$. This means we can first eliminate self crossings of $L$ and then we may deal with the crossings between the components. Is this possible always?
Q2: Can we first apply a finite sequence of Reidemeister moves to split $L_1$ from the link $L$ without affecting the crossings which are not between $L_1$ and $L_i, i=1,2,3$. If so, then we may split the components of the link first and then deal with the self crossings of each component.
I know that the answer of both of my questions is no if the link $L$ is not trivial link. The whitehead link with five crossings is a counterexample. How about the case of an unlink?