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From Hatcher's Algebraic Topology ( available here ) Page 22 :

Adding oriented circles and deforming to get the following links:

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.

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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 $8$.

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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 $\mathbb{A}$.

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Showing the Same , with the Dark Green & light Green lines indicating the Corresponding Points moving around.

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    $\begingroup$ What I'm concerned about is tracking the point $x_0$. Since both curves intersect $x_0$ and approach it from two directions, how do you separate the two loops (touching at $x_0$) without cutting or introducing a discontinuity at $x_0$. The criticality of singular points like these have shown up when proving, for instance, that a line is not homeomorphic to a circle. $\endgroup$
    – Nate
    Jul 25, 2023 at 9:39
  • $\begingroup$ Core Point here is that the given Curve B is going above & below the Curve A , the Crossings will indicate that we are in 3D. In Case 1 , Curve is Crossing itself in 3D & not Cutting itself. In Case 2 , Curve B is touching itself & not Cutting itself. Crossing & touching are OK in 3D. What is not allowed is changing "Crossing from above" to "Crossing from below" , which will require Cutting. $\endgroup$
    – Prem
    Jul 25, 2023 at 10:53
  • $\begingroup$ I have two more thoughts to high-light [[1]] Hatcher Explicitly talks about allowing that Crossing Point $x_0$ [[ "This is one reason why loops are defined merely as continuous paths, which are allowed to pass through the same point many times" ]] [[2]] All this is in the Context of ADDITION of loops , which Implicitly involves cutting two loops at $x_0$ & joining back to get a larger loop though $x_0$. That will give the "Group Structure" of ADDITION of loops. $\endgroup$
    – Prem
    Jul 25, 2023 at 11:39
  • $\begingroup$ I see, so if you can cut and rejoin you're allowed to glue back together, and this guarantees that the bijection/isomorphism holds, since every point in the curves are preserved? $\endgroup$
    – Nate
    Jul 25, 2023 at 11:57
  • $\begingroup$ In the Context of Hatcher making "Group Structure" of ADDITION of loops , that is Correct , though it is allowed ONLY at the Point $x_0$ ! $\endgroup$
    – Prem
    Jul 25, 2023 at 12:08

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