Can we modify one component of a link and keep the others unchanged

Question

Suppose a link diagram consists of two components $K_1$ and $K_2$. Suppose $K_1$ has some self crossings as well as some crossings with the other component $K_2$. If $K_1$ is isotopic to the trivial knot (unknot), my question is can we transform $K_1$ to the trivial knot by applying Reidemeister moves to the self crossings only and keep the crossings with $K_2$ unchanged? In general, can we just modify one component and keep the other unchanged.

Answer 1

If I interpreted your question correctly, then no, it is not possible. The following link is called the Whitehead link.

Let's say that the orange component is $K_1$ and the blue component is $K_2$. The component $K_1$ has one self-crossing, the component $K_2$ has no self-crossings, and there are four crossings between $K_1$ and $K_2$.

If we were able to "untie" $K_1$ by performing Reidemeister moves only involving self-crossings of $K_1$, then the resulting link would only have four crossings: $0$ self-crossings of any kind and $4$ crossings between $K_1$ and $K_2$. The Whitehead link is alternating (just look at the picture), and Tait's conjecture (proven by Kauffman, Murasugi, and Thistlethwaite) states that any reduced alternating diagram minimizes crossing number. Thus there are no diagrams of the Whitehead link with fewer than 5 crossings.