Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret subgroup of $PSL(2,\mathbb{C})$ and also $\pi_1(M)$ is isomorphic to $\Gamma$.

Most of the hyperbolic knots I have seen above $\Gamma$ are always generated by parabolic elements, like 4_1 knot, twist knots, etc.. So, natural questions arise

My question is

1. Does there exist a hyperbolic knot such that $\Gamma$ is two generated Kleinian group where the generators are not commutative and not parabolics?
2. More generally does there exist a cusp hyperbolic $3-$ manifold $M=\mathbb{H}^3/\Gamma$ where two elements and the generators generate $\Gamma$ are not commute each other and they also not parabolic elements?

My attempted,

I think the answer is yes for 2nd one,

I am taking two loxodromic elements say $\gamma_1$ and $\gamma_2$ such that whose geodesic axis $l_{\gamma_1}$ and $l_{\gamma_2}$ passing from their fixed points far(they don't share the fixed points) from each other, I believe that $\Gamma= <\gamma_1,\gamma_2>$ from a Kleinian group.

Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret subgroup of $PSL(2,\mathbb{C})$ and also $\pi_1(M)$ is isomorphic to $\Gamma$.

Most of the hyperbolic knots I have seen above $\Gamma$ are always generated by parabolic elements, like 4_1 knot, twist knots, etc.. So, natural questions arise

My question is

1. Does there exist a hyperbolic knot such that $\Gamma$ is two generated Kleinian group where the generators are not commutative and not parabolics?
2. More generally does there exist a cusp hyperbolic $3-$ manifold $M=\mathbb{H}^3/\Gamma$ where $\Gamma$ is Kleinian group of two generators, generators of $\Gamma$ are not commute each other and they also not parabolic elements?

My attempted,

I think the answer is yes for 2nd one,

I am taking two loxodromic elements say $\gamma_1$ and $\gamma_2$ such that whose geodesic axis $l_{\gamma_1}$ and $l_{\gamma_2}$ passing from their fixed points far(they don't share the fixed points) from each other, I believe that $\Gamma= <\gamma_1,\gamma_2>$ from a Kleinian group.

Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret subgroup of $PSL(2,\mathbb{C})$ and also $\pi_1(M)$ is isomorphic to $\Gamma$.

Most of the hyperbolic knots I have seen above $\Gamma$ are always generated by parabolic elements, like 4_1 knot, twist knots, etc.. So, natural questions arise

My question is

1. Does there exist a hyperbolic knot such that $\Gamma$ is two generated Kleinian group where the generators are not commutative and not parabolics?
2. More general does there exist a cusp hyperbolic $3-$ manifold $M=\mathbb{H}^3/\Gamma$ where $\Gamma$ is generated by non commutative non parabolic elements?

My attempted,

I think the answer is yes for 2nd one,

I am taking two loxodromic elements say $\gamma_1$ and $\gamma_2$ such that whose geodesic axis $l_{\gamma_1}$ and $l_{\gamma_2}$ passing from their fixed points far(they don't share the fixed points) from each other, I believe that $\Gamma= <\gamma_1,\gamma_2>$ from a Kleinian group.

Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret subgroup of $PSL(2,\mathbb{C})$ and also $\pi_1(M)$ is isomorphic to $\Gamma$.

Most of the hyperbolic knots I have seen above $\Gamma$ are always generated by parabolic elements, like 4_1 knot, twist knots, etc.. So, natural questions arise

My question is

1. Does there exist a hyperbolic knot such that $\Gamma$ is two generated Kleinian group where the generators are not commutative and not parabolics?
2. More generally does there exist a cusp hyperbolic $3-$ manifold $M=\mathbb{H}^3/\Gamma$ where two elements and the generators generate $\Gamma$ are not commute each other and they also not parabolic elements?

My attempted,

I think the answer is yes for 2nd one,

I am taking two loxodromic elements say $\gamma_1$ and $\gamma_2$ such that whose geodesic axis $l_{\gamma_1}$ and $l_{\gamma_2}$ passing from their fixed points far(they don't share the fixed points) from each other, I believe that $\Gamma= <\gamma_1,\gamma_2>$ from a Kleinian group.