Correctness of the Wirtinger presentation

The input code has finite vertex, edge, segment and crossing lists, so the rational-coordinate, incidence, cyclic-order and starting-arc checks are finite. Each pairwise segment test uses terminating rational determinant, intersection and endpoint-order operations. A failed validation returns invalid-input. If validation succeeds, the one-component tracing follows the finite cyclic list of edges and paired half-edges from $a_0$, returns to $a_0$ after finitely many steps, and labels each maximal under-arc once. The crossing pass then visits the finite crossing list once, reads each source and target from its verified cyclic order, appends exactly the corresponding positive-meridian relator, and returns the finite presentation. Thus the procedure terminates with one generator per under-arc and one relator per crossing, as claimed. If $c=0$, the explicit punctured-product model in [F1] gives its fundamental group as infinite cyclic on the positive meridian (the sphere factor is simply connected), while [F6]–[F8] identify the returned one-generator empty-relator presentation with the same free cyclic group. If $c>0$, [F1] and [F5] give the free meridian basis for the meridian region.

Assume $c>0$ and order the crossings $v_1,\dots ,v_c$. Let $b_{\ast }$ be the common collar basepoint in $U$, and let $X_0=U$ and $X_i=U\cup V_1\cup \cdots \cup V_i$. For each $i$, the relative-open cover $X_i=X_{i-1}\cup V_i$ has intersection $A_i=U\cap V_i$, the annulus specified in [F1]. Choose $q_i$ on its core and use the supplied collar access path ${\gamma }_i$ in $U$ from $b_{\ast }$ to $q_i$. Concatenation and reparameterization make $[\alpha ]\mapsto [{\bar{\gamma }}_i\ast \alpha \ast {\gamma }_i]$ a well-defined homomorphism from loops based at $b_{\ast }$ to loops based at $q_i$. Its inverse uses ${\bar{\gamma }}_i$, since the inserted path-reversal pairs contract by the piecewise retracing homotopy. The same concatenation formula commutes with inclusion. Apply [F3] at $q_i$: since $V_i$ is contractible, [F4] identifies ${\pi }_1(X_i,q_i)$ with ${\pi }_1(X_{i-1},q_i)$ modulo the normal closure of the annular-core image. Compute that image in ${\pi }_1(U,b_{\ast })$, before attaching any crossing piece. In the cap model of [F1], push its boundary to the top plate at $z=0$; deleting the two exit disks makes this plate a disk with two holes. Let ${\mu }_s,{\mu }_t$ be the exit meridians, and let $\eta$ run from their basepoints across the plate between the exits, below the over-arc. The oriented knot enters one exit and leaves the other, so their positive meridians have opposite planar orientations. Cutting the two-holed plate along access arcs reads its outer boundary, up to orientation and basing conjugacy, as ${\mu }_s\eta {\mu }_t^{-1}{\eta }^{-1}$. All these paths lie in $A\subset U$. For the common collar paths ${\alpha }_s,{\alpha }_t$, the loop ${\alpha }_s\eta {\alpha }_t^{-1}$ represents $m_o$ in $U$: in the over-normal $(y,z)$ plane it runs from the high collar down the source side, across below the over-arc, and back up the target side, winding once positively. Its straightening to the over-meridian stays above the plate and avoids both exit tubes and the over-tube. This is the same local path calculation as [F2], performed entirely in the upper region, without using an attached cap. Therefore the outer-boundary image is, up to conjugacy and inversion, $m_s{m}_o{m}_t^{-1}{m}_o^{-1}=m_o(m_o^{-1}{m}_s{m}_o{m}_t^{-1})m_o^{-1}$. Its normal closure is exactly that of the constructor's relator, whether or not the understrand orientation follows the source-to-target direction. This computes the attaching word in the free group, rather than inferring it from an equation in the final complement. Distinct $V_i$ are disjoint, so these are precisely the intersections at every stage. Repeating over the finite crossing list kills exactly the normal closure of the constructor's relators.

By [F6] and [F7], the quotient in step 2.1 is the returned presentation group; [F8] identifies the quotient map with the homomorphism sending each symbol to its positive meridian. It is an isomorphism onto the group of the covered exterior by the pushout construction.

For $c=0$, step 1.1 already identifies the returned cyclic presentation with the Euclidean complement group on the positive meridian. For $c>0$, step 3.1 identifies the presentation with the covered exterior group, and the compact-tube retraction identifies that group with ${\pi }_1({\mathbb{R}}^3\setminus K)$. In both cases [F9] then gives the isomorphism to ${\pi }_1(S^3\setminus K)$, with exactly the stated positive-meridian assignment.

By [F2], replacing the common basing path conjugates every generator image by the same element. Simultaneous conjugation is an inner automorphism of the target group, so the resulting presentation group remains isomorphic to ${\pi }_1(S^3\setminus K)$.

Changing the designated start arc reindexes the same finite list of under-arcs. The induced bijection of generator symbols carries each crossing's over/under labels, and therefore each constructor relator, to the corresponding relator in the new indexing. It extends to an isomorphism of the free groups and carries one relator normal closure onto the other; [F7] and [F8] give an isomorphism of the presented quotients.