Let $T_1$ be the standart trefoil knot, embedded in $\mathbb R^3$. Then, one can easily give a simple Wirtinger presentation of $\pi_1(\mathbb R^3 \setminus T_1)$ by $\langle a,b,c | a = bcb^{-1}, b=cac^{-1}, c=aba^{-1} \rangle$. Killing one generator and substituting the corresponding, superflous relation into the other relators, we get the reduced presentation $\langle a,b| a = baba^{-1}b^{-1}, b= abab^{-1}a^{-1} \rangle$. Knowing that $\pi_1(\mathbb R^3 \setminus T_1)$ is isomorphic to the braid group $B_3$, I want to simplify it even further, so that $\pi_1(\mathbb R^3 \setminus T_1) = \langle x,y|x^3 = y^2 \rangle$. Now two obvious candidates are $x = ba$ and $y= bab$. Unsing the above relations, one can immediately compute that $a = yx^{-1}$ and $x^3=y^2$. However, I was so far unable to express $b$ in terms of $x$ and $y$. Can anybody help me out here, it might be something really obvious I am missing....
Further, let $T_1,....T_n$ be a collection of disjoint trefoils with $n \geq 2$ in $\mathbb R^3$ that also have disjoint images under a regular projection of the corresponding link. Then, one can construct the connected sum $T := T_1 \# .....\# T_n$, which is now again a knot in $\mathbb R^3$. Then, one can compute a group presentation of $\pi_1(\mathbb R^3 \setminus T)$ in terms of $n + 1$ generators and $n$ relators, given by $\langle x_i,y| x_i^3 = (x_i x_1^{-1}yx_1^{-1} x_i)^2\rangle $ for $i=1,..n$.
Question: Does there exist a presentation of $\pi_1(\mathbb R^3 \setminus T)$ with less than $n+1$ generators?
EDIT: I found a somewhat simpler presentation, given by $\langle a,b_i: ab_ia = b_iab_i \rangle$ for $i=1,...,n$. Is it clear that this presentation cannot be reduced further ?
It might very well be that a solution to the first question is very helpful for answering the second one. Any help is appreciated
