“…The value of N (g, h) has already been computed except N (g, 0) for g ≥ 3, N (g, 1) for g ≥ 2 and N (2,2). Recently Baykur and Korkmaz [1] found an interesting relation in the mapping class group M 1 2 of the genus-2 surface with one boundary component, and by using this relation they showed that N (2, 0) = 7. Furthermore, using the 8-holed torus relation [15] and the Matsumoto-Cadavid-Korkmaz relation [3,13,19], Hamada [12] found an upper bound for N (g, 1): he showed that N (g, 1) ≤ 4 if g ≥ 5 and N (g, 1) ≤ 6 if g = 3, 4.…”