You might be familiar with how a torus can be thought of as identifying sides of a square. If not, look at this little gif from wikipedia: Identification square. A torus knot can be though of as a line with rational slope that passes through the square and when you get to an edge, you move to the opposite edge and continue on at the same slope.
So, a $(1,1) Torus knot is the diagonal of the square, which looks like this.
But if you were to take a cylinder and stick a knife through it, you would cut it in two places.
If you were to follow the line as it cuts through the torus, or for us, the cylinder (we are cutting it before it would be bend around to make the usual torus) , you will see we get two independent lines that intersect the cylinder. One that starts at the bottom and one that starts at the top, which the second picture shows in red and black, respectively.
So, the corresponding identification square should have two $(1,1)$ torus knots, one in black and one in red.
Hope this was helpful.