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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

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  • $\begingroup$ When you say 3-equivalent, do you mean you have to use exactly 3 k-moves or less than or equal to 3 k-moves? $\endgroup$ – Osama Ghani Jul 16 '17 at 7:14
  • $\begingroup$ Two knots are 3-equivalent if there is a series of 3-moves which can take us from projection of the first knot to the second. $\endgroup$ – Ajay Kumar Nair Jul 17 '17 at 1:32
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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

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