In a previous question I asked about a specific fundamental group problem, which was resolved via SVK but I was also interested in whether or not the Wirtinger presentation was valid in some way. In that case, my space was a finite graph and it was answered positively that the Wirtinger presentation does extend to the complement of a finite graph
More generally, to which situations does the Wirtinger presentation extend?
For example, suppose I want to calculate the fundamental group of $\mathbb{R}^3 - X$ where $X$ is the union of the $x$-axis, $y$-axis and the curve $$t\mapsto (\cos(t), \sin(t), \cos(2t))$$ which is a circle that wraps under the $x$-axis and above the $y$-axis. I computed the fundamental group using SVK and with the Wirtinger presentation (treating the intersection of the axes at the orgina as both an over and under crossing) and got the same answer. Is this a valid procedure or a coincidence?
This example is some sort of mixing of a graph and a link or knot, does the Wirtinger presentation still extend to this example?
Is the following picture, taken from Topology and Groupoids, of an embedding of a 1-complex $K^1$ into $\mathbb R^3$, general enough for you?
You could add some infinite lines.
As background theory you should really have the groupoid, i.e. many base point, version of the Seifert-van Kampen Theorem, which I published in 1967, and developed in the 1968, 1988 edition of the above book (and is not in any other topology text in English!). For an example of an application, see this recent preprint, which gives a useful result on a pushout of not necessarily connected groupoids. The reason is that you think of the above picture as in a slab in $X= \mathbb R^3 \backslash K^1$ dividing 3-space into two halves and write $X$ as $X_1 \cup X_2$ two open sets whose intersection is non connected. (Details in T&G.) So you start with many base points for the transition from topology to algebra, in this case the algebra of groupoids, for which see also the downloadable Categories and Groupoids.