Lower-central series and Magnus expansion for links

For any group $G$, define its lower-central series by ${\mathrm{\Gamma }}_{1}G=G$ and ${\mathrm{\Gamma }}_{q+1}G=[G,{\mathrm{\Gamma }}_{q}G]$, where $[G,H]$ is the subgroup generated by $gh{g}^{-1}{h}^{-1}$ for $g\in G$ and $h\in H$. For $r\ge 1$, let $F_r=⟨x_1,\dots ,x_r⟩$ be the free group on the formal meridian symbols $x_i$, with the free-group convention of its stated generators.

Let ${\mathcal{A}}_r=\mathbb{Z}⟨\text{\hspace{0.17em}⁣}⟨X_1,\dots ,X_r⟩\text{\hspace{0.17em}⁣}⟩$ be the ring of formal integer sums ${\sum }_w{a}_w{X}_w$ over all finite words $w$ in the letters $X_i$, with multiplication given by concatenation of words. This is the degree-complete ring for the word-length filtration; a series with invertible constant coefficient is a unit, with inverse determined recursively degree by degree. Filter it by total degree (word length), and let $\mathfrak{m}$ be the ideal of series with zero constant term. The Magnus expansion is the group homomorphism $E:F_r⟶\mathbb{Z}⟨\text{\hspace{0.17em}⁣}⟨X_1,\dots ,X_r⟩\text{\hspace{0.17em}⁣}{⟩}^{\times },E(x_i)=1+X_i\mathrm{.}$ It sends $x_i^{-1}$ to $(1+X_i{)}^{-1}={\sum }_{k\ge 0}(-1{)}^k{X}_i^k$. For an integer $d\ge 0$, calculations through degree $d$ use the truncated ring $R_d={\mathcal{A}}_r\mathrm{/}{\mathfrak{m}}^{d+1}$. As a $\mathbb{Z}$-module it has the finite monomial basis of words of length at most $d$. In $R_d$, the inverse of $1+A$ for $A\in \mathfrak{m}$ is the finite sum $1-A+A^2-\cdots +(-A{)}^d$.

For a based string link, $x_1,\dots ,x_r$ may be used as formal symbols corresponding to the positive bottom-end meridians. No freeness or lower-central quotient isomorphism is asserted here; those properties require separate proofs.

Free group on a declared generator set

Based endpoint meridians and basings

The commutator convention is $[a,b]=ab{a}^{-1}{b}^{-1}$. Truncation to degree $d$ retains the constant and homogeneous terms of degrees at most $d$.