Non-additivity for triple point numbers on the connected sum of surface-knots

Any surface-knot F in 4-space can be projected into 3-space with a finite number of triple points, and its triple point number, t(F), is defined similarly to the crossing number of a classical knot. By definition, we have t(F-1#F-2) less than or equal to t(F-1) + t(F-2) for the connected sum. In this paper, we give infinitely many pairs of surface-knots for which this equality does not hold.