TITLE: Is a "knot knot" or "double knot" a thing in knot theory?
QUESTION [6 upvotes]: I apologize in advance for my rudimentary knowledge of knot theory, but I've been trying to find out about the significance (if any) of taking a knot (particularly a torus knot), cutting it, and forming another knot using it.  Some examples:
trefoil knot of a trefoil knot

and another "double trefoil" knot

Is there a generally accepted name for this?  Is the resulting knot prime? Are there any other generalizations one can make about these? 
Thank you for any information you have on this.

REPLY [8 votes]: A satellite knot (or link) is more general that your construction.  To make a satellite knot, you take a knot in a solid torus (not in a ball within the solid torus), and embed the solid torus so that it is knotted. 
A doubled knot (or Whitehead double of a knot) is special type of satellite knot that is different from yours. 
Your knots are cablings of a knot, where the knot in the solid torus is a torus knot following a curve on the boundary. The slope is usually parametrized $(p,q)$ and yours are $(2,n)$ cablings for some $n$. Nontrivial cablings are prime (see Burde and Zieschang, Knots, p. 93) so your knots are prime.

Thanks to Ryan Budney for pointing out that my initial definition of satellite know was more restricted than a more common one.