http://www.varf.ru/rudn/manturov/book.pdf
I am reading p.56 in the book (p.69 in the pdf file), and trying to understand the proof that quandles completely determine knots up to orientation.
The only thing I understand is that it tries to show that $ana^{-1} = an^{-1}a^{-1}$ which would mean $n = n^{-1}$ and since $n$ is an element in the fundamental group of a torus, isomorphic to $\mathbb{Z}^2$, this means that $2n = (0,0)$, i.e. $n =(0, 0)$. After some rereading, I find that the book seems to point to the fact that the normalizer of the meridian is isomorphic to the fundamental group of the torus $\partial N(K)$. But why is it so?
An explanation that aids understanding of the proof would be very much appreciated.
See page 52 of Joyce's 1979 dissertation An Algebraic Approach to Symmetry with Applications to Knot Theory http://aleph0.clarku.edu/~djoyce/quandles/aaatswatkt.pdf