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 ) , l ≡ 1 , 2 ( mod 6 ) , and T ( 4 , 5 ) . Also, for any positive integer k we construct infinitely many 3-braid knots with alternation number k.