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.
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