A k-move is defined to be a local change in a knot projection that replaces two untwisted strings with two strings that twist around each other with $k$ crossings in a right-handed manner. A -k-move is the same with left-handed twists.
We say that two knots are k-equivalent if we get from one projection of a knot to a projection of the otherthrough a series of k-moves or -k-moves.
Now I need to prove that the three knots in the image below are 3-equivalent to a trivial link. I think that first k-move or -k-move on the trivial link would always result in trefoil. I couldn't proceed further.
The braid group on two strands is isomorphic to $\mathbb{Z}$, and 3-moves let you work instead in $\mathbb{Z}/3\mathbb{Z}$. In a knot diagram, this means a sequence of two right/left-hand crossings can be replaced with a single left/right-hand crossing (respectively), in addition to the ability to introduce or eliminate sequences of three crossings.
In the following picture, $\sim$ is knot equivalence and $\sim^\text{3-move}$ is 3-equivalence.