On Torus Knots obtained by Self-Fusions on the Torus

For the torus knot on the standard torus in S3, we can get another torus knot by performing a self-fusion on the torus. In the present paper, we detect the relationship between those two torus knots, and show a necessary and sufficient condition for the torus knot to become the trivial knot by once self-fusion. Moreover for a given positive integer p > 0, we show a necessary and sufficient condition for T(p, q) to become the trivial knot by maximal times of self-fusions.