I'm thinking yes, because they are both a quotient of the square. But I can't figure out what the actual homeomorphism is. Do we have to "go outside of $\mathbb{R}^3$" with the homeomorphism?
ADDITION: is there an ambient isotopy between them?
I'm thinking yes, because they are both a quotient of the square. But I can't figure out what the actual homeomorphism is. Do we have to "go outside of $\mathbb{R}^3$" with the homeomorphism?
ADDITION: is there an ambient isotopy between them?
Do we have to "go outside of $\mathbb R^3$" with the homeomorphism?
Exactly. There is no homeomorphism of $\mathbb R^3$ sending one to the other (look at the boundary: in one case it is two parallel circles but in the other the circles are intertwined, even if that's not (yet) a formal proof), but as abstract topological space, they are homeomorphic.
I'm thinking yes, because they are both a quotient of the square.
Exactly. You just have to convince yourself that the identification is the same. You then have an explicit homeomorphism.