Wirtinger presentation correctness for labeled links

If a component has no underpassing point, insert a small type-I curl in a disk meeting only that component. The local curl is the boundary of a disk pushed into a ball disjoint from the rest of the link, so removing it by the reverse isotopy identifies the resulting link with the original. After this finite operation each component has an underpass. Apply [F5] to obtain the normalized height model required by the cover supplier. If $h$ is the resulting ambient isotopy's endpoint, transport each specified tube parametrization by $h$, together with the basepoint and all based paths. Then $h(E_L)$ is exactly the exterior of these transported tubes, and $h$ identifies its based positive meridians with the original ones. Work in this normalized model and retain the names $L,E_L$.

By [A1, F1], the upper-region group is free on $x_1,\dots ,x_n$, the lower-region group is trivial, and the overlap group is normally generated by the crossing circles $c_1,\dots ,c_n$. Applying [F2] to the open cover of $X$, the pushout with a trivial lower group is the quotient of the upper group by the normal closure of the images of these circles. This quotient initially computes ${\pi }_1(X)$. Use common basepoint-change paths when passing between the overlap basepoint and the reference upper basepoint; this conjugates all generator images by the same path, consistently with the given common basing.

By [F3], the normal closure in the upper group of the images of the crossing circles is the normal closure of the relators $r_c$. By [F4], the pushout quotient is therefore the stated Wirtinger presentation of ${\pi }_1(X)$. Now apply the radial retraction from [F1]: in each punctured product tube, write a point as $(\theta ,r\omega )$ with $0<r\le 1$ and $\mathrm{\mid }\omega \mathrm{\mid }=1$, and use $(\theta ,r\omega )\mapsto (\theta ,((1-t)r+t)\omega )$ for $0\le t\le 1$, fixing the exterior. These formulas agree with the identity at $r=1$, so they give the based deformation retraction $R:X\to E_L$. It sends a positive small normal circle to the corresponding positive tube-boundary meridian, and fixes any already exterior-based loop. Thus composing the presentation isomorphism with $R_{\ast }$ gives exactly the stated isomorphism to ${\pi }_1(E_L)$ with its positive-meridian marking; retracting the common basepoint-change paths gives the same based identification. Transporting back by $h^{-1}$ restores the originally specified exterior.

The cover lemma also states that any $n-1$ of the crossing circles normally generate the overlap group. Replacing the full list by any list omitting one circle therefore gives the same normal closure in the upper group, hence the same group ${\pi }_1(X)$ and, under the fixed isomorphism $R_{\ast }$, the same marked exterior-group presentation from step 3.1.

For an arbitrary access $a_e^{\mathrm{\prime }}$ to the same positive meridian circle reached by the reference access $a_e$, set $g_e=a_e^{\mathrm{\prime }}{a}_e^{-1}$. Cancellation of reversed paths gives $a_e^{\mathrm{\prime }}{\mu }_e(a_e^{\mathrm{\prime }}{)}^{-1}=g_e(a_e{\mu }_e{a}_e^{-1})g_e^{-1}=g_e{x}_e{g}_e^{-1}$ in the actual exterior group and hence, by step 3.1, in its reference presentation. If one common based loop is prepended to all accesses, this is simultaneous conjugation and so an inner automorphism of the target. Independent $g_e$ need not define an automorphism or even a generating tuple, which is why no such assertion is made. All geometric based meridian data are retained by these explicit words.