Upper bound for the alternation number of a torus knot

摘要

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.