Expanded rotatope (InstanceTopic, 3)
From Hi.gher. Space
The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope. For example, the torus is homeomorphic to the duocylinder, so the duocylinder is the expanded rotatope of the torus. The expanded rotatope is always embedded in a higher dimension than the toratope, but they have the same minimal frame.
To find the expanded rotatope of a toratope written in toratopic notation, look at each group, including nested groups, and write down the number of elements in the group. Then append any outer digons to the result.
- ((II)I) => 22, since there are two groups each with two elements.
- ((III)I) => 32, since there is one group with three elements, and another group with two elements (the sub-group counts as one element).
- ((II)II) => 32
- ((II)I)I => 221, since there are two groups with two elements, and an extra digon on the outside.
- ((II)(II)) => 222, since there are two (II) groups, and another group containing these two groups.
- (((II)II)(II)) => 3222
Expanded rotatopes can be used to find homeomorphisms between toratopes. For example, since (22) and ((21)1) have the same expanded rotatope, 222, we have (22) ~ 222 ~ ((21)1)
If two shapes are homeomorphic then they have the same homology groups. It follows that to calculate homology groups for all toratopes, we need only calculate them for all rotatopes.