How does the one-point compactification affect the fundamental group of the complement of a knot?

Question

I've just begun to study knot theory and I'm wondering how does the one-point compactification of $\mathbb{R}^3$ change the knot group of a knot $K$, e.g. if $\pi(\mathbb{R}^3 \backslash K)$ is isomorphic to $\pi(\mathbb{S}^3 \backslash K)$. My intuition is that yes, they're isomorphic, but how could I proceed the formal proof?

Thank you in advance for your help.

Answer 1

Let $\infty$ denote the point added to $\mathbb{R}^3$ to make $S^3$, and let $U$ be a contractible open neighbourhood of $\infty$ in $S^3\setminus K$. Then $S^3\setminus K = (\mathbb{R}^3\setminus K)\cup U$. Now apply the Seifert-van Kampen Theorem.