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In light of Knot groups with big number of generators, I was wondering...

Question 1 What is the minimal number of generators of the fundamental group of a satellite knot?

Another more specific question

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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    $\begingroup$ @MarkSapir All knots have abelianized fundamental group isomorphic to $\mathbb Z$, so you won't get any nontrivial bounds the way you suggest. The linked thread discusses some nontrivial results for knot groups. $\endgroup$
    – Wojowu
    Commented Jul 17, 2021 at 14:04
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    $\begingroup$ In general there aren’t good results known for ranks of amalgams. A very basic question might be: must the rank of a non-trivial satellite knot be at least 3? $\endgroup$
    – HJRW
    Commented Jul 17, 2021 at 21:11

1 Answer 1

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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 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.

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