The calculation of the fundamental group of a $(m, n)$ torus knot $K$ is usually done using Seifert-Van Kampen theorem, splitting $\mathbb{R}^3\backslash K$ into a open solid torus (with fundamental group $\mathbb{Z}$) and its complementary (with fundamental group $\mathbb{Z}$). To use Seifert-Van Kampen properly, usually the knot is thickened so that the two open sets overlap in a path-connected open set with fundamental group $\mathbb{Z}$. And the resulting fundamental group is $\langle x, y \ | \ x^m = y^n\rangle$.
I find the thickening part a bit artificial. My question is: can we save this proof avoiding the thickening? For example, does it exist a version of Seifert-Van Kampen for which we can use the closed solid torus (minus the knot) and the closure of its complementary?
Best Answer
To be a knot in the first place, it essentially needs a well-defined regular neighborhood, sort of as proof of its tameness. If you think of a knot as being in the one-skeleton of a triangulation of $\mathbb{R}^3$, then if you want a triangulation of the complement without too many more simplices, you can remove a regular neighborhood of the knot -- after all, this is a deformation retract of $\mathbb{R}^3-K$. It is also possible to triangulate $\mathbb{R}^3-K$, but it takes infinitely many more simplices!
In any case, the van Kampen theorem applies to pairs of subcomplexes whose intersection is a path connected subcomplex. The proof involves taking a neighborhood of the intersection that deformation retracts onto the intersection, where the neighborhood is formed from neighborhoods close to faces of incident simplices, then adding this neighborhood to the pair of subcomplexes. (This is very similar to how subcomplexes of a complex form a "good pair," in Hatcher's terminology.)
There is a cell structure of the torus minus $K$ with infinitely many cells, and one can extend this to a cell structure of $\mathbb{R}^3-K$. If $X_1,X_2$ are closed sets with $X_1\cup X_2=\mathbb{R}^3$ and $X_1\cap X_2$ being the torus, then $X_1-K$ and $X_2-K$ inherit the cell structure, and the van Kampen theorem for complexes applies.
(Note: if you took the solid torus minus $K$ and the closure of the complement of this, as you suggest, their union would be all of $\mathbb{R}^3$! This is like the example of covering $S^1$ by $[0,1/2]$ and $(1/2,1]$ through the quotient map $[0,1]\to S^1$. These are two simply connected subsets of $S^1$ that intersect at a point, but van Kampen, if it were to apply, would give $\pi_1(S^1)=1$. This illustrates what open sets are meant to handle, but also $(1/2,1]$ is not a subcomplex of $S^1$.)