Is there a nontrivial oriented link with two components with linking number 0 which is related to the unknot by a single skein relation? While thinking about a problem of determining whether a given link is a slice link or not, I was lead to the following question: Is there an oriented link with two components (other than the unlink) 1. whose linking number is zero 2. and which, after modifying the link by applying skein relation at some place once, becomes the unknot (with a single component)? * * * **Edit:** I found one by myself and posted it as an answer.

I found one by myself: ![enter image description here](

Here the three tiwsts have an effect of cancelling out the -3 linking number made from the double trefoil.

Similarly, every 2-component link with genus 1 can be realized in this way, and we can always cancel out its linking number by giving some number of additional twists.