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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.

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  • $\begingroup$ If $K_1$ is the trivial knot, then what does it mean to transform it into the trivial knot? Do you want to perform Reidemeister moves on $K_1$ self-crossings only until $K_1$ has no more self-crossings? $\endgroup$ Jan 17, 2018 at 11:30

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

enter image description here

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.

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  • $\begingroup$ I mean apply Reidemeister moves to only self crossings of $K_1$ to untie it. So the Whitehead link is a counterexample. I see. Thank you so much. $\endgroup$
    – user113715
    Jan 17, 2018 at 18:06
  • $\begingroup$ I see that the Whitehead link has non-trivial coloring. What if the link admit only trivial coloring for any coloring by a qaundle. Is there also a counterexample in this case? $\endgroup$
    – user113715
    Jan 17, 2018 at 19:52
  • $\begingroup$ Sorry, I know very little about quandle colorings of links. Maybe someone else can chime in. $\endgroup$ Jan 17, 2018 at 21:49
  • $\begingroup$ I mean is it also true if the link is unlink, in other words $K_1$ and $K_2$ can be splitted and each is an unknot. $\endgroup$
    – user113715
    Jan 20, 2018 at 20:22

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