Is a "Twisted Torus" its Own Topological Shape?

Question

Assume you have a torus. Cut it at some place to make a cylinder. Twist one end of the cylinder 360 degrees. Glue the ends back together. Is this "twisted" torus different topologically than a regular torus?

Answer 1

Here is my way to look at it intuitively :

CASE 1 :
We can take a sheet with 4 sides (Dimension : $1$ Unit $\times$ $1$ Unit) and stick or glue a Pair of Opposite Sides to get a Cylinder.
Then we can stick or glue the other Pair of Opposite Sides (1A) to get a torus.

Having the Cylinder , we can stick or glue after giving a twist of $180^0$ (1B) or a twist of $360^0$ (1C) , we will still get the torus.

CASE 2 :
With a Variation :
When we have the Cylinder , we can Draw a Black line on the sheet , top to bottom , then we can Draw a Blue line ( Parallelly ) with Distance $1/2$ Unit between the 2 lines.

Now , we can stick or glue the Pair of Opposites sides (2A) with no twist , we will get the same torus with the Black line looping around AND the Blue line looping around.
We have 2 untangled lines here !

Having the Cylinder , if we stick or glue after giving a twist of $180^0$ (2B) we will see that the Black line merges with (connects to) the Blue line which then merges with (connects to) the Black line.
We have a Single line !

Back to the Cylinder , if we stick or glue after giving a twist of $360^0$ (2C) we will see that the Black line merges with itself but it is winding around the torus. Likewise, the Blue line will merge with itself after winding around.
We have 2 tangled lines here !

In All Cases , (1A) (1B) (1C) (2A) (2B) (2C) , we get the torus but the lines are not matching in (2B) & (2C) !

We can see that with "the line Diagrams" :

(1A) (1B) (1C) with no lines :

(2A) no twist :

(2B) $180^0$ twist :

(2C) $360^0$ twist :

Here is a Course & a Paper [ material for reference ] to make sense of "the line Diagrams" :
https://oertx.highered.texas.gov/courseware/lesson/3362/overview
https://mathcircle.berkeley.edu/sites/default/files/handouts/2018/topology_of_surfaces_2018_0.pdf