Return to Answer

Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=1$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce a closed 3-manifold. Use sphere recognition algorithms to check if the result of Dehn filling is homeomorphic to the 3-sphere.

--- Edit ---

As emphasized by the comments of Agol and Igor Rivin, this is not an algorithm until one specifies exactly how to compute which Dehn filling curves in $\partial M$ need to be checked. I'll say how to do this when the monodromy $\phi : S \to S$ is pseudo-Anosov, which corresponds to checking whether $M$ is a fibered hyperbolic knot complement (something similar will work to check whether $M$ is a satellite knot whose torus decomposition has a hyperbolic component incident to $\partial M$; something different needs to be done when $M$ is Seifert fibered or is a satellite knot complement whose torus decomposition has a Seifert fibered component incident to $\partial M$).

First compute the longitude and the degeneracy curves $\ell,\delta \in H_1(\partial M;\mathbb{Z})$.

The longitude $\ell$ is the generator of the kernel of $H_1(\partial M;\mathbb{Z}) \to H_1(M;\mathbb{Z}) \approx \mathbb{Z}$.

To compute $\delta$, first compute an invariant train track $\tau \subset S$ such that some component $C$ of $S \setminus \tau$ is a crown surface containing $\partial S$; one can use standard train track algorithms to do this, such as the Bestvina-Handel algorithm. Then suspend $\tau$ to get a branched surface $B$ in $M$. On $B$, take the suspension curve of a cusp of $C$. Isotope that curve through an annulus$\times [0,1]$ component of $M \setminus B$ to get $\delta \subset \partial M$. From a more invariant but less algorithmic point of view, if one suspends the stable geodesic measured lamination $\lambda \subset S$ of $\phi$ to get an essential lamination $\Lambda$ in $M$, the curve $\delta$ is the unique one which is isotopic to a curve in a leaf of $\Lambda$.

Now compute the three curves on $\partial M$ whose geometric intersection with $\ell$ equals $1$ and whose geometric intersection with $\delta$ equals $0$ or $1$. Those are the only three possible curves along which Dehn filling can give a 3-sphere: every Dehn filling along a curve which intersects $\ell$ either $0$ times or $\ge 2$ times has nontrivial 1st homology with $\mathbb{Z}$ coefficients; and every Dehn filling along a curve which intersects $\delta$ more than $1$ time has an essential lamination.

Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=\mathbb{Z}$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce a closed 3-manifold. Use sphere recognition algorithms to check if the result of Dehn filling is homeomorphic to the 3-sphere.

--- Edit ---

As emphasized by the comments of Agol and Igor Rivin, this is not an algorithm until one specifies exactly how to compute which Dehn filling curves in $\partial M$ need to be checked. I'll say how to do this when the monodromy $\phi : S \to S$ is pseudo-Anosov, which corresponds to checking whether $M$ is a fibered hyperbolic knot complement (something similar will work to check whether $M$ is a satellite knot whose torus decomposition has a hyperbolic component incident to $\partial M$; something different needs to be done when $M$ is Seifert fibered or is a satellite knot complement whose torus decomposition has a Seifert fibered component incident to $\partial M$).

First compute the longitude and the degeneracy curves $\ell,\delta \in H_1(\partial M;\mathbb{Z})$.

The longitude $\ell$ is the generator of the kernel of $H_1(\partial M;\mathbb{Z}) \to H_1(M;\mathbb{Z}) \approx \mathbb{Z}$.

To compute $\delta$, first compute an invariant train track $\tau \subset S$ such that some component $C$ of $S \setminus \tau$ is a crown surface containing $\partial S$; one can use standard train track algorithms to do this, such as the Bestvina-Handel algorithm. Then suspend $\tau$ to get a branched surface $B$ in $M$. On $B$, take the suspension curve of a cusp of $C$. Isotope that curve through an annulus$\times [0,1]$ component of $M \setminus B$ to get $\delta \subset \partial M$. From a more invariant but less algorithmic point of view, if one suspends the stable geodesic measured lamination $\lambda \subset S$ of $\phi$ to get an essential lamination $\Lambda$ in $M$, the curve $\delta$ is the unique one which is isotopic to a curve in a leaf of $\Lambda$.

Now compute the three curves on $\partial M$ whose geometric intersection with $\ell$ equals $1$ and whose geometric intersection with $\delta$ equals $0$ or $1$. Those are the only three possible curves along which Dehn filling can give a 3-sphere: every Dehn filling along a curve which intersects $\ell$ either $0$ times or $\ge 2$ times has nontrivial 1st homology with $\mathbb{Z}$ coefficients; and every Dehn filling along a curve which intersects $\delta$ more than $1$ time has an essential lamination.

Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=1$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce a closed 3-manifold. Use sphere recognition algorithms to check if the result of Dehn filling is homeomorphic to the 3-sphere.

--- Edit ---

As emphasized by the comments of Agol and Igor Rivin, this is not an algorithm until one specifies exactly how to compute which Dehn filling curves in $\partial M$ need to be checked. I'll say how to do this when the monodromy $\phi : S \to S$ is pseudo-Anosov, which corresponds to checking whether $M$ is a fibered hyperbolic knot complement (something similar will work to check whether $M$ is a satellite knot whose torus decomposition has a hyperbolic component incident to $\partial M$; something different needs to be done when $M$ is Seifert fibered or is a satellite knot complement whose torus decomposition has a Seifert fibered component incident to $\partial M$).

First compute the longitude and the degeneracy curves $\ell,\delta \in H_1(\partial M;\mathbb{Z})$.

The longitude $\ell$ is the generator of the kernel of $H_1(\partial M;\mathbb{Z}) \to H_1(M;\mathbb{Z}) \approx \mathbb{Z}$.

To compute $\delta$, first compute an invariant train track $\tau \subset S$ such that some component $C$ of $S \setminus \tau$ is a crown surface containing $\partial S$; one can use standard train track algorithms to do this, such as the Bestvina-Handel algorithm. Then suspend $\tau$ to get a branched surface $B$ in $S$. On $B$, take the suspension curve of a cusp of $C$. Isotope that curve through an annulus$\times [0,1]$ component of $M \setminus B$ to get $\delta \subset \partial M$. From a more invariant but less algorithmic point of view, if one suspends the stable geodesic measured lamination $\lambda \subset S$ of $\phi$ to get an essential lamination $\Lambda$ in $M$, the curve $\delta$ is the unique one which is isotopic to a curve in a leaf of $\Lambda$.

Now compute the three curves on $\partial M$ whose geometric intersection with $\ell$ equals $1$ and whose geometric intersection with $\delta$ equals $0$ or $1$. Those are the only three possible curves along which Dehn filling can give a 3-sphere: every Dehn filling along a curve which intersects $\ell$ either $0$ times or $\ge 2$ times has nontrivial 1st homology with $\mathbb{Z}$ coefficients; and every Dehn filling along a curve which intersects $\delta$ more than $1$ time has an essential lamination.

Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=1$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce a closed 3-manifold. Use sphere recognition algorithms to check if the result of Dehn filling is homeomorphic to the 3-sphere.

--- Edit ---

As emphasized by the comments of Agol and Igor Rivin, this is not an algorithm until one specifies exactly how to compute which Dehn filling curves in $\partial M$ need to be checked. I'll say how to do this when the monodromy $\phi : S \to S$ is pseudo-Anosov, which corresponds to checking whether $M$ is a fibered hyperbolic knot complement (something similar will work to check whether $M$ is a satellite knot whose torus decomposition has a hyperbolic component incident to $\partial M$; something different needs to be done when $M$ is Seifert fibered or is a satellite knot complement whose torus decomposition has a Seifert fibered component incident to $\partial M$).

First compute the longitude and the degeneracy curves $\ell,\delta \in H_1(\partial M;\mathbb{Z})$.

The longitude $\ell$ is the generator of the kernel of $H_1(\partial M;\mathbb{Z}) \to H_1(M;\mathbb{Z}) \approx \mathbb{Z}$.

To compute $\delta$, first compute an invariant train track $\tau \subset S$ such that some component $C$ of $S \setminus \tau$ is a crown surface containing $\partial S$; one can use standard train track algorithms to do this, such as the Bestvina-Handel algorithm. Then suspend $\tau$ to get a branched surface $B$ in $M$. On $B$, take the suspension curve of a cusp of $C$. Isotope that curve through an annulus$\times [0,1]$ component of $M \setminus B$ to get $\delta \subset \partial M$. From a more invariant but less algorithmic point of view, if one suspends the stable geodesic measured lamination $\lambda \subset S$ of $\phi$ to get an essential lamination $\Lambda$ in $M$, the curve $\delta$ is the unique one which is isotopic to a curve in a leaf of $\Lambda$.

Now compute the three curves on $\partial M$ whose geometric intersection with $\ell$ equals $1$ and whose geometric intersection with $\delta$ equals $0$ or $1$. Those are the only three possible curves along which Dehn filling can give a 3-sphere: every Dehn filling along a curve which intersects $\ell$ either $0$ times or $\ge 2$ times has nontrivial 1st homology with $\mathbb{Z}$ coefficients; and every Dehn filling along a curve which intersects $\delta$ more than $1$ time has an essential lamination.

Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=1$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce a closed 3-manifold. Use sphere recognition algorithms to check if the result of Dehn filling is homeomorphic to the 3-sphere.

--- Edit ---

As emphasized by the comments of Agol and Igor Rivin, this is not an algorithm until one specifies exactly how to compute which Dehn filling curves in $\partial M$ need to be checked. I'll say how to do this when the monodromy $\phi : S \to S$ is pseudo-Anosov, which corresponds to checking whether $M$ is a fibered hyperbolic knot complement (something similar will work to check whether $M$ is a satellite knot whose torus decomposition has a hyperbolic component incident to $\partial M$; something different needs to be done when $M$ is Seifert fibered or is a satellite knot complement whose torus decomposition has a Seifert fibered component incident to $\partial M$).

First compute the longitude and the degeneracy curves $\ell,\delta \in H_1(\partial M;\mathbb{Z})$.

The longitude $\ell$ is the generator of the kernel of $H_1(\partial M;\mathbb{Z}) \to H_1(M;\mathbb{Z}) \approx \mathbb{Z}$.

To compute $\delta$, first compute an invariant train track $\tau \subset S$ such that some component $C$ of $S \setminus \tau$ is a crown surface containing $\partial S$; one can use standard train track algorithms to do this, such as the Bestvina-Handel algorithm. Then suspend $\tau$ to get a branched surface $B$ in $S$. On $B$, take the suspension curve of a cusp of $C$. Isotope that curve through an annulus$\times [0,1]$ component of $M \setminus B$ to get $\delta \subset \partial M$. From a more invariant but less algorithmic point of view, if one suspends the stable geodesic measured lamination $\lambda \subset S$ of $\phi$ to get an essential lamination $\Lambda$ in $M$, the curve $\delta$ is the unique one which is isotopic to a curve in a leaf of $\Lambda$.

Now compute the three curves on $\partial M$ whose geometric intersection with $\ell$ equals $1$ and whose geometric intersection with $\delta$ equals $0$ or $1$. Those are the only three possible curves whose Dehn fillings can be 3-spheres: every Dehn filling which intersects $\ell$ either $0$ times or $\ge 2$ times has nontrivial 1st homology with $\mathbb{Z}$ coefficients; and every Dehn filling which intersects $\delta$ more than $1$ time has an essential lamination.

Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=1$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce a closed 3-manifold. Use sphere recognition algorithms to check if the result of Dehn filling is homeomorphic to the 3-sphere.

--- Edit ---

As emphasized by the comments of Agol and Igor Rivin, this is not an algorithm until one specifies exactly how to compute which Dehn filling curves in $\partial M$ need to be checked. I'll say how to do this when the monodromy $\phi : S \to S$ is pseudo-Anosov, which corresponds to checking whether $M$ is a fibered hyperbolic knot complement (something similar will work to check whether $M$ is a satellite knot whose torus decomposition has a hyperbolic component incident to $\partial M$; something different needs to be done when $M$ is Seifert fibered or is a satellite knot complement whose torus decomposition has a Seifert fibered component incident to $\partial M$).

First compute the longitude and the degeneracy curves $\ell,\delta \in H_1(\partial M;\mathbb{Z})$.

The longitude $\ell$ is the generator of the kernel of $H_1(\partial M;\mathbb{Z}) \to H_1(M;\mathbb{Z}) \approx \mathbb{Z}$.

To compute $\delta$, first compute an invariant train track $\tau \subset S$ such that some component $C$ of $S \setminus \tau$ is a crown surface containing $\partial S$; one can use standard train track algorithms to do this, such as the Bestvina-Handel algorithm. Then suspend $\tau$ to get a branched surface $B$ in $S$. On $B$, take the suspension curve of a cusp of $C$. Isotope that curve through an annulus$\times [0,1]$ component of $M \setminus B$ to get $\delta \subset \partial M$. From a more invariant but less algorithmic point of view, if one suspends the stable geodesic measured lamination $\lambda \subset S$ of $\phi$ to get an essential lamination $\Lambda$ in $M$, the curve $\delta$ is the unique one which is isotopic to a curve in a leaf of $\Lambda$.

Now compute the three curves on $\partial M$ whose geometric intersection with $\ell$ equals $1$ and whose geometric intersection with $\delta$ equals $0$ or $1$. Those are the only three possible curves along which Dehn filling can give a 3-sphere: every Dehn filling along a curve which intersects $\ell$ either $0$ times or $\ge 2$ times has nontrivial 1st homology with $\mathbb{Z}$ coefficients; and every Dehn filling along a curve which intersects $\delta$ more than $1$ time has an essential lamination.