From Hatcher's Algebraic Topology ( available here ) Page 22 :
I don't understand how he manages to get a continuous deformation into the loops without "cutting" some things at point $x_0$ for both of the examples (and thus violating the rules of a homeomorphism). He mentions needing to pass the "string" through itself to get these, but I would like to see a sequence of intermediate steps. A drawing for either would be nice.
2 Ways to Visualize that Intuitively :
Visualization 1 :
Imagine that the "Circle" $\mathbb{A}$ is not there.
We can "open" up the Curve $\mathbb{B}$ to look like the Digit $8$.
When the upper Part of the $8$ has clock-wise orientation , the lower Part orientation may have either clock-wise orientation or anti-clock-wise orientation.
We can Draw the $8$ with 2 Curves , which is not what we want here : It will be like cutting the Curve into 2.
We want to Draw the $8$ with a Single Curve.
Hence we see 2 ways to get that :
Case 1 : In left Case , from top , we draw the upper Part (it is clock-wise here) , then move to "Centre" & then to the left (Purple line) , then downward (it is anti-clock-wise here) then move upward , then move to the "Centre" , then move left (Blue line) & reach the Starting Point at the top.
This way has a Crossing along the Dotted line.
Case 2 : In right Case , from top , we draw the upper Part (it is clock-wise here) , then move to the "Centre" & stay on the right (Purple line) , then downward (it is clock-wise here too) then move upward , then move to the "Centre" , then stay on the left (Blue line) & reach the Starting Point at the top.
This way has no Crossing along the Dotted line.
Putting back the "Circle" $\mathbb{A}$ , shown in Grey , we see that the left curve $\mathbb{B}$ is interlocked with $\mathbb{A}$ , while we will see that the right curve $\mathbb{B}$ will not be interlocked with $\mathbb{A}$.
Visualization 2 :
Imagine that we have a "Circle" $\mathbb{B}$ like shown in the left Diagram. We are ignoring $\mathbb{A}$ for the moment.
We can take the top Part & Bottom Part to be made of (Black) rigid wood which we can not bend.
The Middle Parts are made of (Blue & Grey) non-rigid rope which we can bend.
Case 1 : In Middle Diagram , when we move the 2 rope Parts towards each other like shown in the Purple lines with arrows (which are Pointing left & right in $X$ Axis) , the Parts will touch. We can Draw this "Circle" with clock-wise orientation. Both upper Part & lower Part will have Same Orientation.
There will be no Crossing in the Middle.
Case 2 : In right Diagram , we imagine lifting the lower wood Part out of $XY$ Plane , into 3D , rotating it like shown in the Purple lines with arrows (which are Pointing up & down in $Z$ Axis) & then keeping back in $XY$ Plane.
We can Draw this "Circle" with clock-wise orientation in upper Part & anti-clock-wise orientation in lower Part.
There will be a Crossing in the Middle.
Putting back the "Circle" $\mathbb{A}$ , we will see that the Middle curve $\mathbb{B}$ will not be interlocked with $\mathbb{A}$ , while we will see that the right curve $\mathbb{B}$ will be interlocked with $\mathbb{A}$.
Showing the Same , with the Dark Green & light Green lines indicating the Corresponding Points moving around.