what is a delta move on a trefoil knot

Question

I saw some notes discuss delta move, but they just give part of the graph. What is it like if we do the delta move on a trefoil knot? Is it an unknot?

Answer 1

It depends a lot on which position your trefoil is in when you perform the move. Just like a crossing change or any other surgery-related change depends on where you perform it.

However, if you are thinking of a trefoil in the classical position with three crossings, then yes, the result is an unknot (you can unknot it by three Reidemeister I moves).

Using this you can now see how the result could be different: since the Delta move is local, you can perform it on a factor in a connected sum. If you perform it on the second factor in the sum $$3_1\sharp \ U$$ where $U$ is an unknot in a position as described above, you can obtain $$3_1\sharp\ 3_1$$ that is a sum of two trefoils, as the result of a Delta move on one trefoil.

Answer 2

See figure 1 at Uchida, Y., On delta-unknotting operation, Osaka J. of Math., v. 30 (1993), p. 753.

Note that this is an operation on three crossings forming the vertices of a triangular region (thus, "$\Delta$-move") having three strands forming its edges. The pair-wise over- and under-crossings are unchanged before and after the move, but the order of the pairwise crossings is reversed. Each strand enters and leaves the move at the same positions on the boundary of the move before and after the move.

It is easy enough to finish the cited image into a trefoil by adding the same arcs to the left and right images. If, on both diagrams, you complete the loops in the upper-left, upper-right, and bottom, you have a trefoil on the left and an unknot on the right (barely obscured by three type-I Reidemeister twists). If you complete the loops on the top, bottom-left, and bottom-right, the unknot appears on the left and the trefoil on the right. (The two trefoils are homeomorphic via a rotation by $\pi/3$, highlighting that this move does not invert the over-under crossings.)