10

Takashi SHIOYA

a heart not containing the previous Bjs) bounded by subarcs of a would impose the

existence of linking pairs of double points.

1.7. Definition. The semi-regularity index ind(a) = N U {»} of a semi-

regular curve a is the number of disks Z?/ associated to a which are not lemons.

With such a definition we obviously have ind(a) n(a) and ind(a) = 1 if

n(a) = 1.

Remark. Since for a nonsimple semi-regular curve a the disk B\ exists and is

not a lemon, the curve a is simple if and only if ind(a) = 0.

1.8. The sequence of hearts of a semi-regular curve. To a semi-regular

curve a is associated an increasing sequence of integers k(j) such that 0 j

ind(a) defined as follows.

(i) *(0) := 0.

(ii) &(1) := 1 if ind(a) 1, or equivalently if n(a) 1.

(iii) k(j + 1) := min{ / e N ; i k(j) + 1 and£;is a heart }.

1.9. Proposition. Let a be a semi-regular curve and {/(/)} the infinite

sequence of integers defined as follows.