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The $T$-genus of a knot is the minimal number of borromean-type triple pointson a normal singular disk with no clasp bounded by the knot; it is an upperbound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the sliceknots 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 thatthe difference between the $T$-genus and the slice genus can be arbitrarilylarge. We introduce the ribbon counterpart of the $T$-genus and prove that itis 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 coincidefor all knots if and only if all slice knots are ribbon. We work in the moregeneral setting of algebraically split links and we also discuss the case ofcolored links. Finally, we express Milnor's triple linking number of analgebraically split $3$-component link as the algebraic intersection number ofthree immersed disks bounded by the three components.

Slice genus, $T$-genus and $4$-dimensional clasp number