A torus is a surface shaped like a donut. Imagine that I've painted two rings on a torus. One is on the outer surface, and goes through the hole in the center, coming around from the other side. The other ring is on the inner surface, and goes all the way around the hole in the center. These two rings of paint are linked.
I then cut a small hole in the torus. Through this hole, I turn the torus inside-out.
In the process, the rings of paint switch from the outer surface to the inner surface and vice versa. Therefore, they have become unlinked. How?
(In reply to Trying to visualise this- SPOILER
by Sir Percivale)
I have been trying to visualise the turning insideout of a torus using pen and paper. My initial doodles looked like spaghetti junctions. I hope someone can understand what I am about to type. kjkjregek (joke)
I think the main difficulty comes from the relativly complex 3D curvature of a torus so it helps to find a simple 2D representation...
(Draw this) A side-on Xsection of a tori is traditionaly drawn as two circles, side by side.
Lets squash the hole down to the size of a thin straw and distort the surface a bit...
Draw a circle. Now add a vertical diameter. See how this is topologically the same as the above drawing, as if the two circles had been squashed together, so the line down the center of the circle represents a thin straw. Add a dot on the right hand side of the circle to represent the hole we are going to turn this surface through.
Now start to pull that straw through the hole...
Draw a circle again. Now, from the top of the circle draw a line going down, curving to the right, passing through the hole, curving back around (this step isnt meant to be presice, but make sure the line dosent double back on it's self cos that confuses me), passing back through the hole and curving back down to the bottom the circle.
Now pull the right hand side of the spherical part over to the extreme left. (This stage may require some effort in practice, when working with inner-tubes)
Draw the left hand half of the circle as before. Draw the central tube as before. (The entire right hand hemisphere is now pulled over to the left of everything else) Draw a great arc, starting at the top, and curving around the left hand side (almost parallel to the left semi circle) and rejoining at the bottom.
At this stage my drawing looked like a crescent moon with a wiggley line joining the points...
Remember that wiggly line is a tube. Draw the above again with the tube represented as two lines and the two halves of the crescent not quite meeting at the points. You should now have two non-intersecting loops.
Now we pull the shape into a traditional torus form...
Finally draw two concentric circles.
Take two different colour crayons and on each of the above drawings add in thoose paint rings...
We now see there was no paradox here, sure the rings have switched surfaces, but they have also switched location, hence remaining interlinked.