# A ribbon presentation for a torus knot

Let $$K$$ be a knot in $$S^3$$. It is well-known that the knot $$K \# -\overline{K}$$ is always ribbon.

The following picture describes the connected sum of the left-handed torus knot $$T(3,4)$$ and the right-handed torus knot $$T(3,4)$$. In Rolfsen's notation, $$T(3,4) = 8_{19}$$, for several descriptions see  and .

I would like to find the ribbon move(s) for this composite knot but I could not elaborate.

Is there an easy way to see this or any trick to figure out the necessary ribbon moves? If you want to find a set of ribbon moves, then you need to end up with an unlink with several components; therefore, you need to attack crossings! Then I add two bands along twists of given torus knots: Next, we have a link with three components: If you apply Reidemeister moves, you may eventually find an unlink with three components: There is no specific trick for any torus knot (my opinion) but you can control the number of additional ribbon moves.

Assume that $$p$$ and $$q$$ are relatively coprime integers with $$p < q$$. Then for the corresponding ribbon knot $$T(p,q) \# \overline{T(p,q)}$$, the number of ribbon bands is $$p-1$$.

As this composite knot is an example of a symmetric union of knots I suggest that you read my paper "The search for nonsymmetric ribbon knots" (see for instance Figures 1 and 4 and the explanation in the text).