Fundamental group of $\mathbb{R}^3$ minus trefoil knot

Let $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van Kampen theorem or some deformation retract of $X$. I don’t need the answer because a presentation of this group is given by this wiki page I don’t know how to choose an open cover of $X$ or how to deforming $X$ to a more suitable space.

Any help is appreciated.


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One nice method is to think of the trefoil as sitting on the surface of a torus. Think of it as slightly thickened. Then subdivide your space into a slight thickening of the torus (minus the thickened knot) and a slight thickening of the complement. The fundamental groups of each piece are isomorphic to $\mathbb Z$, while the intersection deformation retracts to the torus minus the trefoil, which is an annulus, so also has fundamental group $\mathbb Z$. If you take the loop generating the fundamental group of the annulus and push it into the torus, it winds around three times. If you push it out it winds around twice. So Van Kampen gives the following presentation $\langle x,y\,|\, x^3=y^2\rangle$.

The idea is that you label the arcs of the knot in order. Then at each crossing you get a relation. And your fundamental group is the labels of the arcs and the relations you find. For example, if you have a crossing with labels $a$ for the over arc and $b$ and $c$ for the under arcs, the relation should look something like $$aba^{-1}c=1.$$ Now, be careful, the order matters, but it is explained in this pdf rather well. Also, you can always ignore exactly one of the relations you get. It will be a consequence of the others.