I have a question about a line in the proof of
3.4 Theorem about Wirtinger presentation (Chapter 3.B, page 34) of the knot
group
$G(K)= \pi_1(\overline{S^3-K})$ of a knot $K \subset S^3$ from "Knots" by
Burde, Zieschang and Heusener. Here the link: https://www.degruyter.com/document/doi/10.1515/9783110270785/html?lang=de
The claim is that $G$ has presentation
$$ G=\langle s_1,..., s_n \ \vert \ r_1,..., r_n \rangle $$
where the generators $s_i$ are introduced and discussed in subchapter 3.B and $r_{i;j,k}:= s_js_i^{-\nu_j}s_k^{-1}s_i^{\nu_j}$, for $\nu_j= \pm1$.
The proof considers $\mathbb{R}^3$ as a simplicial complex $\Sigma$
containing $Z$ (The
"projecting cylinder" introduced also in the beginning of 3.B) as a
subcomplex,
and let $\Sigma^*$
be the dual cell complex.
Let now $\omega $ be a contractible curve in
$C:=\overline{S^3-K}$ ,
starting at a vertex $P$ of $\Sigma^*$.
Then it says that by simplicial approximation $\omega$ can be replaced by a path in the 1-skeleton of $\Sigma^*$ and the contractible homotopy by a series of homotopy moves which replace arcs on the boundary of 2-cells $ \sigma^2$ of $\Sigma^*$ by the inverse of the rest.
I not understand this last line that by application of simplicial approximation the contractible homotopy can be replaced by a series of homotopy moves which replace arcs on the boundary of $2$-cells of the dual cell complex $\Sigma^*$ by the inverse of the rest.
What is here meant by the "inverse of the rest" and why simplicial approximation permits such transformations? Could somebody elaborate how these used homotopy moves work in detail?