Which satellite knots are prime?
I do know that connected sum of knots is a satellite operation, but I found this statement:
"the satellite knots all have structures which are well known and documented..., and their primality follows from a simple geometric argument"
(from Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell. 20, 33-48, Fall 1998.)
Presumably they mean that all satellite knots that are not connected sums are prime?
The answer that I was seeking is at Peter Cromwell's Knots and Links, pp 84-85.
Theorem 4.4.1: A proper satellite is prime if its pattern is prime or the trivial knot.
Here the word proper means that at least two strands are involved in the satellite construction (possibly with winding number 0)