Whitehead double I get confused when checking the definition of Whitehead double. In this page, a Whitehead double is a special type of satellite knot. 
But how to understand the sentence "Take a Whitehead double of any knot K"? Is "Whitehead double also an operation on a knot K"?
 A: It is an operation on knots, maybe a little hard to understand from the wikipedia article.  This picture may help.  In the picture you can see one component (J) which is clearly unknotted, and another component (K') which looks like it might self- link at first glance but actually does not.  Hopefully it's not too hard to see that the second component fits inside a nice torus $K' \subset V$ (if it helps, V is a thickening of another knot J' for which $J \cup J'$ is a Hopf link).
To construct a general whitehead double, let Y be your knot of interest, and thicken it to a tubular neighborhood U.  Now choose an (untwisted) embedding f:V->U; the image f(K') is the (untwisted) Whitehead double of Y.  Example 2 in the wikipedia article should show how it looks like when Y is the figure eight knot.
It sort of looks like you split Y into two copies, and at some point you cut the two copies and tie them back up together such that the result is connected, but in the specific way as described by the above procedure.
