Truncated Milnor invariants from a based link diagram

Assume AC. The input is either (a) a based $r$-component string-link diagram with its endpoint collars and product-tube parallels equal to the diagram's blackboard push-offs, or (b) a based finite regular diagram of a closed $r$-component labeled oriented link with chosen product tubes, one initial strand on each component, and a common basepoint and access paths. In either mode require $r\ge 1$. The finite input records its compatible common-upper-collar reference accesses. In the compatible branch the actual accesses are these, up to a common base-path change, and the parallels are the blackboard parallels already specified. In the arbitrary-choice branch it additionally records finite words in the compatible strand meridians for each access difference $g_e$, the parallel endpoint differences $h_{0,j},h_{1,j}$, and its relative meridian winding $k_j$. These are word certificates for the chosen paths, part of the supplied based-word data; no evaluation of an unencoded smooth path is claimed. In closed-link mode the preferred longitude uses its own recorded access $h_j$, which may differ from the initial-meridian access $g_j$. Arbitrary parallels are allowed in this branch, with the recorded $k_j$. The input also includes a maximum index length $N\ge 2$. The output is the integer coefficients ${\mu }_{\tau }(I)$ in string-link mode, or the based coefficients ${\mu }_L^b(I)$, their lower indeterminacy data ${\mathrm{\Delta }}_I^b$, and residues ${\bar{\mu }}_L^b(I)$ in closed-link mode, for every index sequence $I$ with $2\le \mathrm{\mid }I\mathrm{\mid }\le N$.

Throughout this procedure, a strand means a maximal Wirtinger arc between successive underpassing points, or between an endpoint and the nearest underpassing point in string-link mode. Obtain these arcs by joining the diagram edges straight through every overcrossing. A closed component without underpasses has one cyclic strand; a string component without underpasses has one endpoint-to-endpoint strand. Before step 1, identify all compatible edge-meridian symbols on each such arc with its one strand symbol, and make this substitution in every supplied word certificate. Record the chosen initial point on its resulting strand. All series in step 2 are assigned to these arcs, so crossing an overpass retains the same meridian series.

Set $d=N-1$ and work in the truncated noncommutative ring $R_d=\mathbb{Z}⟨X_1,\dots ,X_r⟩\mathrm{/}{\mathfrak{m}}^{d+1},$ where $\mathfrak{m}$ is the ideal generated by the letters. Carry out the following finite procedure.

1. Verify the finite crossing, orientation, over/under, and component data. If a closed-link component has no underpassing point, choose a crossing-free disk meeting only a short arc of that component and avoiding its chosen initial meridian point. Insert a small type-I curl there by the supported disk isotopy and record the resulting based diagram. Transport the basepoint, meridian circles, access paths, product tubes, parallels, and all supplied access, endpoint, and longitude word certificates through this isotopy. Before the curl, the component is one cyclic strand; the new underpass cuts it into one interval strand containing the chosen initial point. Identify that interval's reference meridian symbol with the transported old symbol. Every other strand is carried bijectively by the isotopy, with its reference symbol identified in the same way. In the finite certificates, apply this symbol substitution to every word; no unencoded path evaluation or additional certificate is required. Give every resulting strand $a$ the component label $j(a)$.

2. First compute in the compatible reference root symbols $x_j$, irrespective of the actual access choices. For every strand $a$, initialize its meridian series by $Y_a^{(1)}=1+X_{j(a)}$. For each degree $\mathrm{\ell }=2,\dots ,d$, copy every $Y_a^{(\mathrm{\ell }-1)}$ into the constant and lower-degree part of $Y_a^{(\mathrm{\ell })}$, with new degree-$\mathrm{\ell }$ terms initially zero, and reset each chosen root strand to $1+X_j$. Then do one pass over each component in its oriented order. In closed-link mode start at the chosen initial strand and omit the final undercrossing back to that strand; in string-link mode start at the bottom endpoint and use every undercrossing through to the top endpoint. At an undercrossing with over-strand $o$ and source and target under-arcs $s,t$, use the already available $Y_o^{(\mathrm{\ell }-1)}$ and the current $Y_s^{(\mathrm{\ell })}$ to set $Y_t^{(\mathrm{\ell })}={\mathrm{trunc}}_{\mathrm{\ell }}((Y_o^{(\mathrm{\ell }-1)}{)}^{-1}{Y}_s^{(\mathrm{\ell })}{Y}_o^{(\mathrm{\ell }-1)})$ when the traversal is source-to-target, and set $Y_s^{(\mathrm{\ell })}={\mathrm{trunc}}_{\mathrm{\ell }}(Y_o^{(\mathrm{\ell }-1)}{Y}_t^{(\mathrm{\ell })}(Y_o^{(\mathrm{\ell }-1)}{)}^{-1})$ when it is target-to-source. Here ${\mathrm{trunc}}_{\mathrm{\ell }}$ discards monomials of degree greater than $\mathrm{\ell }$, and every inverse is the finite geometric series in the nilpotent ideal of $R_d$.

3. For each component $j$, traverse its undercrossings in order and form $W_j={\prod }_c(Y_{o(c)}^{(d)}{)}^{{ϵ}_c}$, where the factors are in traversal order and ${ϵ}_c=+1$ for source-to-target and $-1$ for target-to-source. In string-link mode set ${\mathrm{\Lambda }}_j=W_j$. In closed-link mode set ${\mathrm{\Lambda }}_j=(1+X_j{)}^{-{\mathrm{wr}}_j(D)}{W}_j,$ where ${\mathrm{wr}}_j(D)$ is the sum of the determinant signs at self-crossings of component $j$.

4. In the arbitrary-choice branch first use the strand-symbol identification of step 1, then evaluate each supplied transport word by the strand series from step 2. Write $G_e$ for the resulting unit series, and $H_{0,j},H_{1,j}$ for the endpoint transport series. In string-link mode replace the compatible ${\mathrm{\Lambda }}_j=W_j$ by $H_{0,j}(1+X_j{)}^{k_j}{W}_j{H}_{1,j}^{-1}\mathrm{.}$ In closed-link mode replace the compatible preferred series by $H_j(1+X_j{)}^{-{\mathrm{wr}}_j(D)}{W}_j{H}_j^{-1},$ where $H_j=H_{0,j}=H_{1,j}$ is the actual longitude access difference, independently of the initial-meridian difference $G_j$. The arbitrary parallel winding cancels in the preferred correction. All powers and inverses are finite truncated operations. To express this result in the actual component-meridian symbols, form $G_j(1+X_j)G_j^{-1}=1+X_j+R_j(X),\mathrm{deg}{R}_j\ge 2.$ With new letters $V_j$, initialize $Z_j^{(1)}=V_j$ and iterate $Z_j^{(s+1)}={\mathrm{trunc}}_d(V_j-R_j(Z^{(s)})),1\le s<d\mathrm{.}$ Substitute $X_j=Z_j^{(d)}(V)$ in the corrected longitude series. This is its expansion in the chosen actual meridians. For the compatible branch this step is the identity. Rename the final letters $V_j$ as $X_j$.

5. For every $I=(i_1,\dots ,i_k,j)$ with $1\le k\le d$, read the coefficient of $X_{i_1}\cdots X_{i_k}$ in ${\mathrm{\Lambda }}_j$. In string-link mode return this integer. In closed-link mode, also compute the gcd of the coefficients for all proper subsequences of cyclic permutations of $I$ of lengths at least two, with empty gcd equal to zero, and return the corresponding residue class.

All scans, including the supplied transport words and the $d-1$ inverse-substitution iterations, are over finite crossing, strand, degree, and index sets. The procedure makes no polynomial-time claim. If a finite-input validation fails, return invalid-input.

Finite regular diagrams and crossing data

Truncated noncommutative ring and Magnus expansion

Source-to-target meridian substitutions

Peripheral longitude word and self-writhe correction

Integer coefficients and based residues

The per-degree passes use over-strand series only through degree $\mathrm{\ell }-1$: their degree-$\mathrm{\ell }$ terms cancel against the constant term of the conjugated meridian. The output records the based residue data; its independence of closed-link basing is the separate theorem on Milnor invariants.