# Proof of the completeness of knot quandle

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.

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