  The $T$-genus of a knot is the minimal number of borromean-type triple points
on a normal singular disk with no clasp bounded by the knot; it is an upper
bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice
knots by the vanishing of their $T$-genus. We generalize this to provide a
$3$-dimensional characterization of the slice genus. Further, we prove that the
$T$-genus majors the $4$-dimensional positive clasp number and we deduce that
the difference between the $T$-genus and the slice genus can be arbitrarily
large. We introduce the ribbon counterpart of the $T$-genus and prove that it
is an upper bound for the ribbon genus. Interpreting the $T$-genera in terms of
$\Delta$-distance, we show that the $T$-genus and the ribbon $T$-genus coincide
for all knots if and only if all slice knots are ribbon. We work in the more
general setting of algebraically split links and we also discuss the case of
colored links. Finally, we express Milnor's triple linking number of an
algebraically split $3$-component link as the algebraic intersection number of
three immersed disks bounded by the three components.
