I start by saying that I am not an expert in this field and I apologize if the question is too elementary.


Let $K$ be a knot in $S^3$. I denote by $\pi\_1(K)$ the knot group, which is the fundamental group of its exterior:
$$ \pi\_1(K) = \pi\_1(S^3 \smallsetminus K) .$$


The *minimal number of generators* of a knot $K$ is the minimal number of generators of $\pi\_1(K)$.


I am searching for knots with an arbitrarily high minimal number of generators. In particular:


1. I found in <https://arxiv.org/abs/1007.3175> , Lemma 4.17, a reference: *Goodrick, R. (1968). Non-simplicially collapsible triangulations of In*. In this article, the author proves that the connected sum of $n$ copies of a two-bridge knot is a $m$-bridge knot with $m$>$n$. A sharper result should hold from *Schultens, Jennifer (2003). Additivity of bridge numbers of knots*. I cannot understand, though, how this should prove the statement. In particular, by [Knot theory question: bridge number vs. min generators of fundamental group of complement](https://mathoverflow.net/questions/103049/knot-theory-question-bridge-number-vs-min-generators-of-fundamental-group-of-c) , this should not imply the result. Maybe I am missing something in the article, I did not go through the details. However, the article is quite dated and there is, hopefully, a simple way to prove this fact nowadays. So the first question is: is there a simple way to prove that there are knots with an arbitrarily high minimal number of generators?
2. The techniques used seem to rely on the connected sum. What if we search for *prime* knots with an arbitrarily high minimal number of generators?
3. What if we search for *hyperbolic* knots with an arbitrarily high minimal number of generators?


Thank you in advance for the attention.



Knot groups with big number of generators
If $\pi\_1(S^3\setminus K)$ has a presentation with $n$ generators then its representation variety $\mathrm{Hom}(\pi\_1(S^3\setminus K),SL\_2(\mathbb{C}))$ is a subvariety of $(SL\_2(\mathbb{C}))^n$, which has complex dimension $3n$, so any component will have complex dimension at most $3n$. So if you want the minimal number of generators to be arbitrarily large, you just have to find knots with high-dimensional representation varieties. (You can replace $SL\_2(\mathbb{C})$ with other groups if you prefer.)


Cooper and Long ("Remarks on the A-polynomial of a knot", section 8) show how to construct hyperbolic knots where there are components of arbitrarily large dimension. Start with some $n$-fold connected sum, and realize it as the closure of a braid $\beta$ such that the union of the closure $\hat\beta$ with its braid axis is hyperbolic. Lift $\hat\beta$ to the $p$-fold cyclic branched cover of $S^3$, branched over the braid axis -- this will again be $S^3$, since the axis is unknotted -- to get a new knot, which is hyperbolic for all large enough $p$. (This knot will be the closure of the braid $\beta^p$.) Then the representation variety of this new knot turns out to have a component of dimension at least $n$ as well.


