k-equivalence of knots 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. _k-move_ 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. Knots

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.

![3-moves on trefoil, figure-eight, and another knot](