TITLE: Comparison between Van Kampen theorem from Allen Hatcher and "Introduction to knot theory".
QUESTION [8 upvotes]: In Allen Hatcher the statement of the theorem is:


 


But in "Introduction to knot theory" by Richard H. Crowell, to speak about Van Kampen theorem he mentioned all the following propositions:



(3.2) If $X_{1}$ and $X_{2}$ are simply-connected, then so is $X = X_{1} \cup X_{2}$.



$$ (3.5) \Bigg\{ {G_{1} = |\textbf{x} : \textbf{r} |_{\phi _{1}},\\ G_{2} = |\textbf{y} : \textbf{s} |_{\phi _{2}},\\ G_{0} = |\textbf{z} : \textbf{t} |_{\phi _{0}}.} $$



But I do not understand why the 2 books have very different approaches to this theorem, I even feel that there are differences between the theorem in both books, could anyone clarify for me at least why "Introduction to knot theory" approahes the theorem in this way? A comparison between both approaches will be greatly appreciated.

REPLY [2 votes]: The reason seems to be that the books were written at different times and with different aims. One is intended, as you see from its title, as a book on knot theory, the other is a book on algebraic topology, more generally. Crowell was the first to publish in 1958 a comprehensible proof of a more general theorem, and gives a proof by direct verification of the universal property. The Preface of a $1967$  book by W.S. Massey stresses the importance of this idea. Van Kampen's 1933 paper is difficult to follow. 
This universal property is not stated in Hatcher's version. The most general result of this type is in this paper  which uses the fundamental groupoid on a set  of base points, and so allows the computation of the fundamental group of the circle, and of a myriad of other spaces; also the proof generalises to higher dimensions,  computing some relative homotopy groups: see this book.
Hope that helps.