the difference between a strip with two end glue together and a strip with two end glue together with $2\pi$ twist

Question

a strip with two end glue together is homeomorphic to a strip with two end glue together with $2\pi$ twist. I want to know why they are different in $R^{3}$ ? I think that there is a relation stronger than homeomorphism, and I notice that the boundary of the first is two separated circle, but the second is two linked circle, what makes these difference? and what area of geometry study these kind problem? is it related with knot theory? I only know some concept in algebraic topology, would someone explain what character can distinguish them?

Answer 1

Botnh your spaces are $X= S^1\times[0,1]$, but you have two non-isotopic embeddings into $\Bbb R^3$, i.e., there is no continuos map $X\times[0,1]\to\Bbb R^3$ such that each $X\times\{t\}\to\Bbb R^3$ is an embedding$^1$ and $X\times\{0\}\to\Bbb R^3$, $X\times\{1\}\to\Bbb R^3$ correspond to the two shapes in question.

$^1$ if we drop this embedding condition, we define the notion of homotopy, which is weaker than isotopy. In fact, the two shapes are homotopic as we can retract each to a simple (knot-free) circle in $\Bbb R^3$.