On Self-fusions of Torus Knots on the Torus

For the torus knot on the standard torus in R3, we can get another torus knot by performing a self-fusion on the torus. In the present paper, we define the self-fusion number sf(p, q) for the torus knot T(p, q) as the minimum number of self-fusions to become the trivial knot from T(p, q), and detect sf(p, q). In particular, we determine the necessary and suffcient condition that sf(p, q) = 2 if p is even. Moreover we show the periodicity of sf(p, q) (mod p).