Aha, now I understand
I'll calculate the holes for all the closed toratopes up to 5D. Open ones aren't worth bothering with as they are the same as their base. Each sequence starts counting from 2, and obviously ends with an infinite number of zeroes.
(II) ~  (circle)
(III) ~ [0, 1] (sphere)
((II)I) ~  (torus)
(IIII) ~ [0, 0, 1] (glome)
((II)II) ~ [1, 1] (toracubinder)
((III)I) ~ [1, 1] (toraspherinder)
((II)(II)) ~  (tiger)
(((II)I)I) ~  (ditorus)
(IIIII) ~ [0, 0, 0, 1] (pentasphere)
((III)II) ~ [0, 2] (toracubspherinder)
((II)III) ~ [1, 0, 1] (toratesserinder)
((IIII)I) ~ [1, 0, 1] (toraglominder)
((III)(II)) ~ [2, 1] (cylspherintigroid)
(((II)I)II) ~ [2, 1] (toracubtorinder)
(((II)II)I) ~ [2, 1] (toracubindric torus)
(((III)I)I) ~ [2, 1] (toraspherindric torus)
((II)(II)I) ~ [2, 1] (toraduocylinderinder)
(((II)I)(II)) ~  (cyltorintigroid)
(((II)(II))I) ~  (tigric torus)
((((II)I)I)I) ~  (tritorus)
As you can see I've ordered them by their sequences. Questions:
1. Is it possible to find a sensible secondary order for closed toratopes with the same hole-sequence?
2. Is it possible to count the number of n-dimensional closed toratopes with a given hole-sequence for each dimension (and therefore check whether any exist)?
3. Is it possible to list all the hole-sequences for each dimension which apply to at least one closed toratope?
4. Is it possible to derive the group notation from the hole-sequence and an additional meaningful property? If so, what property can be used for this purpose?
Some simple observations about construction:
If you construct a n-oprism R from closed toratopic bases P1
, HS(R) = Σ1≤i≤n
) where HS(A) is the hole-sequence of closed toratope A and the sum of sequences is the sequence of sums.
If you then construct a tigroid T from R, HS(T) = HS(R) + K where K is a sequence with a 1 in its n
th entry (counting from 2, of course) and zeroes everywhere else.
If you construct a torus T from a base P, HS(T) = HS(P) + .