$S$ is the boundary of a genus $n$ handlebody in $S^3$. $\{m_1, m_2,..., m_n\}$ is the collection of the meridian circles of $S$; $\{l_1,l_2,...,l_n\}$ is the collection of the longitude circles on $S$. They are both unlinks.

Suppose there are 2 orientation preserving self-homeomorphisms of $S$, $f$ and $g$, such that $\{f(m_1),f(m_2),\dots,f(m_n)\}$ is isotopic to $\{g(m_1),g(m_2),\dots g(m_n)\}$ as links (that is, there is an isotopy $I_t:\sqcup_{i=1}^n S^1\to S^3$ such that $I_0(S^1_i)=f(m_i)$, $I_1(S^1_i)=g(m_i))$ and they have the same framings ($S$ induces a framing on each $f(m_i), g(m_j)$ such that the framing of $f(m_i)$ is taken by the isotopy to the framing of $g(m_i)$). Similarly for $f(l_i)$ and $g(l_i)$. What can I say about $f$ and $g$?

When $n=1$, I think $f$ has to be isotopic to $g$. Have no idea about $n>1$. In the simple case when $\{f(m_i)\}$ and $\{f(l_i)\}$ are unlinks with framing $0$, what is $f$? (I think uniqueness of Heegaard splitting implies $f$ is identity modulo some simple self-homeomorphisms, right?)

Thanks.