You're looking at the right paper. His results apply to a broad class of 3-manifolds, which knot complements happen to be a part of. I don't have the paper here with me but I believe the class was then called "sufficiently large". Which I believe in this case means irreducible and containing an incompressible surface.

edit: Technically what he's describing is a "complete 3-manifold invariant". To turn it into a complete knot invariant you need the following observation. Given a knot complement you can turn it into a knot by filling in the boundary $S^1 \times S^1$ with a $S^1 \times D^2$. To do that you need a gluing map, which amounts to specifying the slope of the $D^2$-factor in $S^1 \times S^1$. So back in Waldhausen's day, given a knot, the invariant of the knot is the knot complement together with the natural filling slope. The knot complement together with the natural filling slope is the complete invariant of the knot (up to mirror inverse). Waldhausen's paper shows you how if you reduce that information to $\pi_1 \partial M \to \pi_1 M$ together with a the filling slope (thought of as an element of $\pi_1 \partial M$), that is also a complete invariant of the knot.

You're looking at the right paper. His results apply to a broad class of 3-manifolds, which knot complements happen to be a part of. I don't have the paper here with me but I believe the class was then called "sufficiently large". Which I believe in this case means irreducible and containing an incompressible surface.

edit: Technically what he's describing is a "complete 3-manifold invariant". To turn it into a complete knot invariant you need the following observation. Given a knot complement you can turn it into a knot (in some $3$-manifold) by filling in the boundary $S^1 \times S^1$ with a $S^1 \times D^2$. To do that you need a gluing map, which amounts to specifying the slope of the $D^2$-factor in $S^1 \times S^1$. Given a knot, the invariant of the knot is the knot complement together with the filling slope that recovers $S^3$. The knot complement together with this natural filling slope is the complete invariant of the knot (up to mirror inverse). Waldhausen's paper shows you how if you reduce that information to $\pi_1 \partial M \to \pi_1 M$ together with a the filling slope (thought of as an element of $\pi_1 \partial M$), that is also a complete invariant of the knot.

You're looking at the right paper. His results apply to a broad class of 3-manifolds, which knot complements happen to be a part of. I don't have the paper here with me but I believe the class was then called "sufficiently large". Which I believe in this case means irreducible and containing an incompressible surface.

You're looking at the right paper. His results apply to a broad class of 3-manifolds, which knot complements happen to be a part of. I don't have the paper here with me but I believe the class was then called "sufficiently large". Which I believe in this case means irreducible and containing an incompressible surface.

edit: Technically what he's describing is a "complete 3-manifold invariant". To turn it into a complete knot invariant you need the following observation. Given a knot complement you can turn it into a knot by filling in the boundary $S^1 \times S^1$ with a $S^1 \times D^2$. To do that you need a gluing map, which amounts to specifying the slope of the $D^2$-factor in $S^1 \times S^1$. So back in Waldhausen's day, given a knot, the invariant of the knot is the knot complement together with the natural filling slope. The knot complement together with the natural filling slope is the complete invariant of the knot (up to mirror inverse). Waldhausen's paper shows you how if you reduce that information to $\pi_1 \partial M \to \pi_1 M$ together with a the filling slope (thought of as an element of $\pi_1 \partial M$), that is also a complete invariant of the knot.