In light of [Knot groups with big number of generators](https://mathoverflow.net/questions/397498/knot-groups-with-big-number-of-generators), I was wondering...



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> **Question 1** What is the minimal number of generators of the fundamental group of a [satellite knot](https://en.wikipedia.org/wiki/Satellite_knot)?
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Another more specific question



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> **Question 2** If we take a twisted Whitehead double of a nontrivial knot, what is the minimal number of generators of the resulting knot group?
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Minimal number of generators of satellite knot groups
Knot groups of satellite knots can have rank 2, and only the trivial knot can have rank 1, so 2 is the minimal possible number of generators.


The knot group of any tunnel number one knot has a presentation with 2 generators and 1 relator. [Morimoto and Sakuma](https://doi.org/10.1007/BF01446565) classified all satellite knots which have tunnel number one. In their notation (see section 1.7 of their article), I believe that the tunnel number one knot $K(8,3;p,q)$ will be some Whitehead double of the $(p,q)$-torus knot. Thus there exist Whitehead doubles of torus knots whose knot groups have rank 2.


