A clear statement is the following:

A compact 3-manifold $M$ is hyperbolic if and only if it has infinite fundamental group and does not contain any essential surface with $\chi \geqslant 0$.

You may remember that by saying that $M$ is hyperbolic unless there is some clear obstruction, and the obstrucion is $\pi_1$ finite or the existence of some surface with $\chi \geqslant 0$. This statement is pretty general and works for any compact 3-manifold with any kind of (possibly empty) boundary, using the appropriate notions of "hyperbolic" and "essential".

• Hyperbolic means that, after removing tori and Klein bottles from $\partial M$, the manifold admits a complete finite-volume metric with constant curvature $-1$ and with geodesic boundary.
• Essential means that the surface is immersed in $M$ with a map that is $\pi_1$-injective and not homotopic to some map whose image is in some component of $\partial M$. In addition, a 2-sphere that does not bound a 3-disc is essential.

More concretely, an orientable manifold $M$ is hyperbolic if and only if one of the following obstructions occurs:

• $\pi_1(M)$ is finite,
• $M$ contains a 2-sphere which does not bound a 3-disc (hence $M$ is reducible)
• $M$ contains an essential 2-disc (hence $M$ is $\partial$-reducible)
• $M$ contains an essential 2-torus whose $\pi_1$ injects but which is not $\partial$-parallel. Note that the 2-torus may be immersed and not embedded: this occurs precisely in the small Seifert spaces
• $M$ contains an essential annulus (for instance in $\Sigma \times [0,1]$ for any surface $\Sigma$)

This characterization implies the one for knot that you mentioned, in virtue of the following:

• every knot complement has infinite $\pi_1(M)$ and is irreducible,
• a knot complement contains an essential 2-disc if and only if it is trivial,
• a knot complement contains an essential annulus or torus if and only if it is not prime, a torus knot, or a satellite.

A clear statement is the following:

A compact 3-manifold $M$ is hyperbolic if and only if it has infinite fundamental group and does not contain any essential surface with $\chi \geqslant 0$.

You may remember that by saying that $M$ is hyperbolic unless there is some clear obstruction, and the obstrucion is $\pi_1$ finite or the existence of some surface with $\chi \geqslant 0$. This statement is pretty general and works for any compact 3-manifold with any kind of (possibly empty) boundary, using the appropriate notions of "hyperbolic" and "essential".

• Hyperbolic means that, after removing tori and Klein bottles from $\partial M$, the manifold admits a complete finite-volume metric with constant curvature $-1$ and with geodesic boundary.
• Essential means that the surface is immersed in $M$ with a map that is $\pi_1$-injective and not homotopic to some map whose image is in some component of $\partial M$. In addition, a 2-sphere that does not bound a 3-disc is essential.

More concretely, an orientable manifold $M$ is hyperbolic unless one of the following obstructions occurs:

• $\pi_1(M)$ is finite,
• $M$ contains a 2-sphere which does not bound a 3-disc (hence $M$ is reducible)
• $M$ contains an essential 2-disc (hence $M$ is $\partial$-reducible)
• $M$ contains an essential 2-torus whose $\pi_1$ injects but which is not $\partial$-parallel. Note that the 2-torus may be immersed and not embedded: this occurs precisely in the small Seifert spaces
• $M$ contains an essential annulus (for instance in $\Sigma \times [0,1]$ for any surface $\Sigma$)

This characterization implies the one for knots that you mentioned, in virtue of the following:

• every knot complement has infinite $\pi_1(M)$ and is irreducible,
• a knot complement contains an essential 2-disc if and only if it is trivial,
• a knot complement contains an essential annulus or torus if and only if it is not prime, a torus knot, or a satellite.

A clear statement is the following:

A compact 3-manifold $M$ is hyperbolic if and only if it has infinite fundamental group and does not contain any essential surface with $\chi \geqslant 0$.

You may remember that by saying that $M$ is hyperbolic unless there is some clear obstruction, and the obstrucion is $\pi_1$ finite or the existence of some surface with $\chi \geqslant 0$. This statement is pretty general and works for any compact 3-manifold with any kind of (possibly empty) boundary, using the appropriate notions of "hyperbolic" and "essential".

• Hyperbolic means that, after removing tori and Klein bottles from $\partial M$, the manifold admits a complete finite-volume metric with constant curvature $-1$ and with geodesic boundary.
• Essential means that the surface is immersed in $M$ with a map that is $\pi_1$-injective and not homotopic to some map whose image is in some component of $\partial M$ or in a ball.

More concretely, an orientable manifold $M$ is hyperbolic if and only if one of the following obstructions occurs:

• $\pi_1(M)$ is finite,
• $M$ contains a 2-sphere which does not bound a 3-disc (hence $M$ is reducible)
• $M$ contains an essential 2-disc (hence $M$ is $\partial$-reducible)
• $M$ contains an essential 2-torus whose $\pi_1$ injects but which is not $\partial$-parallel. Note that the 2-torus may be immersed and not embedded: this occurs precisely in the small Seifert spaces
• $M$ contains an essential annulus (for instance in $\Sigma \times [0,1]$ for any surface $\Sigma$)

This characterization implies the one for knot that you mentioned, in virtue of the following:

• every knot complement has infinite $\pi_1(M)$ and is irreducible,
• a knot complement contains an essential 2-disc if and only if it is trivial,
• a knot complement contains an essential annulus or torus if and only if it is not prime, a torus knot, or a satellite.

A clear statement is the following:

A compact 3-manifold $M$ is hyperbolic if and only if it has infinite fundamental group and does not contain any essential surface with $\chi \geqslant 0$.

You may remember that by saying that $M$ is hyperbolic unless there is some clear obstruction, and the obstrucion is $\pi_1$ finite or the existence of some surface with $\chi \geqslant 0$. This statement is pretty general and works for any compact 3-manifold with any kind of (possibly empty) boundary, using the appropriate notions of "hyperbolic" and "essential".

• Hyperbolic means that, after removing tori and Klein bottles from $\partial M$, the manifold admits a complete finite-volume metric with constant curvature $-1$ and with geodesic boundary.
• Essential means that the surface is immersed in $M$ with a map that is $\pi_1$-injective and not homotopic to some map whose image is in some component of $\partial M$. In addition, a 2-sphere that does not bound a 3-disc is essential.

More concretely, an orientable manifold $M$ is hyperbolic if and only if one of the following obstructions occurs:

• $\pi_1(M)$ is finite,
• $M$ contains a 2-sphere which does not bound a 3-disc (hence $M$ is reducible)
• $M$ contains an essential 2-disc (hence $M$ is $\partial$-reducible)
• $M$ contains an essential 2-torus whose $\pi_1$ injects but which is not $\partial$-parallel. Note that the 2-torus may be immersed and not embedded: this occurs precisely in the small Seifert spaces
• $M$ contains an essential annulus (for instance in $\Sigma \times [0,1]$ for any surface $\Sigma$)

This characterization implies the one for knot that you mentioned, in virtue of the following:

• every knot complement has infinite $\pi_1(M)$ and is irreducible,
• a knot complement contains an essential 2-disc if and only if it is trivial,
• a knot complement contains an essential annulus or torus if and only if it is not prime, a torus knot, or a satellite.