Upper bound for the alternation number of a torus knot

We give an upper bound for the alternation number of a torus knot which is of either 3-, 4-, or 5-braid or of other special types. Using the inequality relating the alternation number, signature, and Rasmussen s-invariant, discovered by Abe, we determine the alternation numbers of the torus knots T(3,l), , and T(4,5). Also, for any positive integer k we construct infinitely many 3-braid knots with alternation number k.