Milnor basing indeterminacy and self-crossing scope

Choose a finite regular diagram of the link. If a component has no underpass, insert a small type-I curl on a short arc in a ball meeting no other component, as in the local disk-push construction of the link-presentation theorem in [F4]. There are finitely many components, so the balls may be chosen disjoint and away from the common basepoint; the supported ambient isotopies carry the original chosen tubes, positive meridians, preferred longitudes and all access paths to their images. On each peripheral torus this transport preserves degree one and zero self-meridian coordinate. The induced based group isomorphism carries every original meridian to its transported meridian, so each original finite free word remains literally the same word and is still a lift of the corresponding transported longitude in every finite nilpotent quotient. This reduction imposes no upper-basing or coherence condition on the original word systems. The augmented diagram now has an underpass on every component, including originally crossingless components; its canonical upper basing will be compared with arbitrary transported basings and lifts by the subsequent variable-substitution and coefficient-ideal argument. First use the geometric root meridians and coherent longitude words $l_a$ from [F5], through degree $k$. For any index sequence $J$, write $c(J)$ for the corresponding coefficient of these words. Let $D_I$ be the ideal of $\mathbb{Z}$ generated by $c(J)$ for all proper subsequences of cyclic permutations of $I$ of length at least two. Its nonnegative generator is the gcd in [F1]. Work in the ring $R_I$ with finitely many surviving monomials, obtained from $(\mathbb{Z}\mathrm{/}{D}_I)⟨X_1,\dots ,X_r⟩$ by setting to zero every monomial whose index word is not a subsequence of $w$. The discarded words span a two-sided ideal: adjoining letters cannot turn a word that is not a subsequence into one that is. Thus multiplication and inversion of series with constant term $1$ are well-defined finite operations. Occurrences, including repeated indices, are retained in their original order; no assumption of distinct indices is used here.

For a self-crossing change on component $a$, quotient the diagram group by the normal closure of all commutators of two conjugates of its meridian, doing this for every component. These quotients on the two sides of the crossing change have identical based presentations. To see this locally, cut both branches into incoming and outgoing arc symbols. The branch that is over has equal incoming and outgoing symbols. On the under-branch the Wirtinger equation conjugates its incoming meridian by the over-meridian; both are conjugates of the same component meridian and commute in this quotient, so its outgoing symbol also equals its incoming symbol. Switching the crossing interchanges which branch uses which argument and leaves precisely the same two equalities. All other crossing relations, and root paths chosen outside this small ball, are unchanged. Thus the identification preserves the component meridians, with no assertion of an isomorphism of the original unreduced groups.

The coefficient of any surviving word in $E([x_a,l_a])-1$ is zero in $R_I$. Indeed put $E(l_a)=1+L_a$. The exact identity is $E([x_a,l_a])-1=(X_a{L}_a-L_a{X}_a)(1+X_a{)}^{-1}(1+L_a{)}^{-1}\mathrm{.}$ Every summand contributing to a surviving word therefore contains one distinguished occurrence of $a$ and a nonempty block from $L_a$, as well as possibly other blocks. If that block has index word $B$, its coefficient is $c(Ba)$. The selected block and the distinguished $a$ occupy disjoint positions in a subsequence of $w$; starting just after that occurrence of $a$ and reading cyclically shows that $Ba$ is a subsequence of a cyclic permutation of $I$. It is proper, since its length is at most $k<\mathrm{\mid }I\mathrm{\mid }$. Hence $c(Ba)\in D_I$, and the whole summand vanishes. The same argument applies if $(1+L_a{)}^{-1}$ is expanded elsewhere in the expression; the distinguished block can always be taken from the displayed first factor. This proves the assertion for each relator, its inverse, conjugates and finite products. Also $E({\mathrm{\Gamma }}_{k+1}F)=1$ through degree $k$ by [F6]. Therefore [F5] makes the coefficient of $X_w$ independent modulo $D_I$ of every lift of a fixed longitude in $G\mathrm{/}{\mathrm{\Gamma }}_{k+1}G$.

Under that reduced-group identification, the longitude of a component different from $a$ is unchanged: its traversal has no undercrossing factor in the altered ball, and its other arc meridians have the common images in step 1.2. For component $a$, the local factor lost or gained is a conjugate of its own meridian. It disappears on further quotienting by the normal subgroup generated by that meridian. The self-writhe normalization also differs by a power of this same meridian, so the preferred longitudes agree in this further quotient. These statements follow from the actual ordered undercrossing product, not solely from the meridian crossing relation.

Consider a different basing. Each new meridian is a conjugate of the old one, and each new longitude is also a conjugate of the old degree-one zero-self-meridian curve. The conjugating loops may be different if the two accesses were chosen independently; retain both recorded loops. Lift the finitely many conjugating loops to $F\mathrm{/}{\mathrm{\Gamma }}_{k+1}F$ by [F4]. The resulting change of root variables is $\varphi (x_a)=u_a^{-1}{x}_a{u}_a$. This is an automorphism of the finite nilpotent free quotient. It is the identity on degree one, hence on every lower-central graded layer, whose commutators are generated by the degree-one generators by [F7]. To construct a preimage, correct a word successively in ${\mathrm{\Gamma }}_s\mathrm{/}{\mathrm{\Gamma }}_{s+1}$ for $s=1,\dots ,k$; each correction has the prescribed image because the graded map is the identity. For injectivity, an element in the kernel is successively in ${\mathrm{\Gamma }}_2,\dots ,{\mathrm{\Gamma }}_{k+1}=1$. Its inverse is again a substitution by conjugates of the same respective generators. Thus the new longitude words are obtained by inverse variable substitution, their own independently recorded longitude conjugations, and elements of the transformed canonical kernel described in [F5]. No assertion that an independently based new meridian and longitude commute is needed; [F5] describes the reference presentation before these separate output conjugations.

Now suppose the entries of $I$ are distinct. In the ring $R_I$ of step 1.1 every monomial containing an index twice is zero. For any word $u$, every term of $E(u^{-1}{x}_{b}u)-1$ contains $b$. Products of two such positive-degree expressions are therefore zero when both refer to $b$, and their units commute. The additional reduced-group relators $[u^{-1}{x}_{b}u,v^{-1}{x}_{b}v]$ are consequently annihilated. By [F5], the kernel of the map to the reduced nilpotent group consists exactly of these additional relators, the peripheral commutators already annihilated in step 2.1, and ${\mathrm{\Gamma }}_{k+1}F$. If the changed component is $j$, the further normal-meridian subgroup in step 2.2 is annihilated too, since $j$ does not occur in $w$. Thus the two longitude coefficients agree modulo $D_I$. If the changed component is not $j$, their equality in the reduced group from step 2.2 already gives this conclusion.

In a variable substitution of this kind, $X_a$ is replaced by $X_a$ plus terms of degree at least two, every one of which contains an occurrence of $a$: this follows by expanding $E(u_a{)}^{-1}(1+X_a)E(u_a)-1$. A coefficient of $X_w$ in the substituted longitude therefore differs from its old coefficient only by integer multiples of $c(Bj)$ for proper subsequences $B$ of $w$. These are generators of $D_I$. A longitude conjugation has the same property by direct multiplication, or by [F2]: its constant term cancels and every correction uses a shorter longitude coefficient. The transformed kernel is also annihilated. In the argument of step 2.1, substituting conjugates preserves the distinguished occurrence of each original variable, so the selected $B$ and the distinguished $a$ still form a proper cyclic subsequence of $I$. Hence every permitted basing and lift change preserves the target coefficient modulo the ideal of the old proper coefficients.

These assertions also identify the modulus exactly, including when it is zero. Induct on index length. At length two there are no lower coefficients; commutator kernels and all conjugation corrections start in degree two, so the degree-one coefficients agree as integers. For a longer $I$, apply the induction hypothesis to every proper cyclic subsequence $J$ of $I$. It says that its coefficient changes by an integer combination of coefficients on proper cyclic subsequences of $J$, all of which are proper cyclic subsequences of $I$. Order these finitely many sequences by length. The new coefficients thus lie in the ideal of the old proper coefficients. Conversely solve the same triangular relations in increasing length to express each old coefficient in the ideal of the new proper coefficients. The two ideals, their nonnegative gcd generators, and hence their residue groups are equal. Step 3.1 then proves equality of the target residues. Independent choices of word lifts at the separate finite degrees in [F1] are covered by step 2.1 and this same triangular argument; no unmentioned coherence hypothesis is imposed on those inputs.

Induct again on the distinct index length. Length-two coefficients are preserved by step 3.2 with zero lower ideal. For larger $I$, apply the lower-length conclusions to its proper cyclic subsequences. The triangular ideal argument of step 5.1 shows that the old and new proper-coefficient ideals are equal; step 3.2 therefore gives equality in the canonically identical residue group. Step 4.1 permits arbitrary basings and finite word lifts at either endpoint. Composing along the stipulated finite sequence proves the self-crossing assertion. For repeated indices the square-zero argument in step 3.2 is unavailable, and no such assertion has been made. This proves both claims with the inherited AC and no extra choice principle.