Diagram word for a preferred peripheral longitude

Set $v_0=1$. Suppose the current strand meridian on $K_j$ is represented by $v_{i-1}^{-1}{x}_j{v}_{i-1}$. At crossing $c_i$, if the component passes source-to-target, [F1] replaces it by $x_{o(c_i)}^{-1}{v}_{i-1}^{-1}{x}_j{v}_{i-1}{x}_{o(c_i)}$, which is $v_i^{-1}{x}_j{v}_i$ for $v_i=v_{i-1}{x}_{o(c_i)}$. If it passes target-to-source, solving the same relation for the source meridian gives the same formula with $v_i=v_{i-1}{x}_{o(c_i)}^{-1}$. Thus after one traversal the ordered product is exactly $W_j(D)$.

In the local product chart at each undercrossing, the narrow band between the component and its parallel has boundary access paths whose difference is the over-meridian factor just computed in step 1.1. Away from crossings the parallel and the core run in a product annulus and contribute no group letter. Concatenating these finitely many local annuli from bottom to top gives the based string-link longitude; concatenating once around a closed component gives its based blackboard-parallel loop. In both cases the ordered word is $W_j(D)$.

In the closed-link case the parallel in step 2.1 has degree $+1$ on $K_j$. By [F2] and [F3], its self-meridian coordinate is the exponent sum of those factors in $W_j(D)$ whose over-strand belongs to $K_j$. Each self-crossing contributes exactly once, at its underpass. Source-to-target means the under-tangent has positive determinant with the over-tangent, so its exponent is the crossing sign $+1$; target-to-source gives both signs $-1$ by [F4]. Consequently the coordinate is exactly ${\mathrm{wr}}_j(D)$. Every mixed-crossing factor lies in a different meridian coordinate and contributes zero to this coordinate. This proves the needed self-writhe correction directly from the actual group word and homology basis, without invoking a countable-choice framing or linking-number theorem.

For arbitrary strand accesses, path concatenation gives $y_e=g_e{x}_e{g}_e^{-1}$ and hence $x_e^{ϵ}=g_e^{-1}{y}_e^{ϵ}{g}_e$ for $ϵ=\pm 1$. Substitution in the finite ordered product proves ${\widehat{W}}_j=W_j$ exactly, without asserting that independent meridian conjugations cancel. A common base-path change conjugates every factor and the entire word by that one path, recovering the literal-word branch with its correspondingly based meridians.

In the closed-link case, multiplication by $x_j^{-{\mathrm{wr}}_j(D)}$ changes the self-meridian coordinate by $-{\mathrm{wr}}_j(D)$ and leaves the degree on $K_j$ equal to $+1$. Its self-meridian coordinate is therefore zero. By [F3], this is the preferred zero-self-meridian longitude class. In the string-link case, step 2.1 already gives the based longitude specified by the tube and access paths.

The tube annulus of a string component is $[0,1]\times S^1$. Two paths between its fixed endpoint coordinates differ, relative endpoints, by the integer winding of their concatenation in $S^1$: lift that coordinate to $\mathbb{R}$ and linearly interpolate paths with the same lifted endpoints. Thus ${\beta }^{\mathrm{\prime }}\simeq {\mu }_0^k\beta$. On a closed component the same calculation on $S^1\times S^1$ gives the unique class $(k,1)$ of a degree-one parallel relative to the blackboard section; its based loop is ${\mu }_0^k\beta$. Concatenating actual accesses now gives $a_0^{\mathrm{\prime }}{\beta }^{\mathrm{\prime }}(a_1^{\mathrm{\prime }}{)}^{-1}=h_0(a_0{\mu }_0^k{a}_0^{-1})(a_0\beta a_1^{-1})h_1^{-1}=h_0{x}_j^k{\widehat{W}}_j{h}_1^{-1}\mathrm{.}$ For a closed component $h_0=h_1=h$ and conjugation changes no homology coordinate. Steps 2.1 and 3.1 give coordinate $k+{\mathrm{wr}}_j(D)$. Multiplication by the actual initial meridian $(h{x}_j{h}^{-1}{)}^{-(k+{\mathrm{wr}}_j(D))}$ cancels both the chosen extra winding and the self-writhe, leaving $h{x}_j^{-{\mathrm{wr}}_j(D)}{\widehat{W}}_j{h}^{-1}$. It has degree one and zero self-coordinate, so [F3] identifies precisely the preferred class. These formulas cover no undercrossings, $k=0$, independently changed endpoints, and common conjugation, and preserve the arbitrary data of the peripheral definition.