Take a toratope A. Replace the numbers with groups of 1s in brackets, e.g. 4 = (1111).

Now for each integer k > 1, count the number of pairs of brackets in A with k objects in them. Call this number A_k. We could call this the number of k-holes in A.

Conjecture 1:

If A_k = B_k for all k > 1, then A is homeomorphic to B. That is, there is a continuous bijective map from A to B with a continuous inverse.

Example: A = (211), B = (31)

Rewrite as A = ((11)11), B = ((111)1)

A_2 = 1, since only the inside bracket has two objects in it.

B_2 = 1 because only the outside bracket has two objects.

A_3 = 1 because only the outside bracket has three objects in it

B_3 = 1 because only the inside bracket has three objects in it

A_4 = B_4 = 0

Here's a list of which toratopes are homeomorphic if my conjecture is true.

(21) ~ 22 (this we already know)

(211) ~ (31) ~ 32 (I've tried to find an actual map between these, so far I haven't been completely successful. However I've shown they have the same homology groups, so they must be homeomorphic)

((21)1) ~ (22) ~ 222 (I had an argument with wendy about this years ago. Turns out she was partially right, the shapes are homeomorphic but they're not exactly the same object.)

(2111) ~ (41) ~ 42 (possibly others)

(221) ~ (32) ~ ((21)11) ~ ((211)1) ~ ((31)1) ~ 322 (quite likely others)

Conjecture 2:

Let A be an n-dimensional toratope (not a toratope embedded in n dimensions). The homology groups of A are as follows

H

_{0}A = Z

H

_{1}A = A

_{2}Z (that is, A

_{2}copies of Z summed together)

H

_{k}A = A

_{k+1}Z for 1 < k < n

H

_{n}A = Z