Tracing and slicing are inverse on relative classes

Trace is well-defined on path classes. Define $T([\alpha ])$ to be the braid-isotopy class of the braid traced by the unique lift of $\alpha$ from $Q$, which exists by [L1]. If $\alpha$ and ${\alpha }^{\mathrm{\prime }}$ represent the same based path-homotopy class by [L5, L6], then [L2] makes their traced braids braid-isotopic. Thus $T$ is independent of the representative.

Slice is well-defined on braid-isotopy classes. Suppose $Z$ is a braid isotopy from $\beta$ to ${\beta }^{\mathrm{\prime }}$. Apply the real-complex identification [L10] to its coordinates. By joint continuity in [L4] and the product criterion [L7], the tuple map $z:I_s\times I_t\to (\mathrm{int}{D}^2{)}^n,z(s,t)=(Z_1(s,t),\dots ,Z_n(s,t))$ is continuous. Each slice is collision-free by [L11, L12], so its image lies in $F_n(\mathrm{int}{D}^2)$; the subspace topology [L8] makes the restricted map continuous. Composing with the continuous orbit projection [L15] gives $H(s,t):=p_n(z(s,t))\in C_n(\mathrm{int}{D}^2)\mathrm{.}$ The fixed bottom tuple gives $H(s,0)=[Q]$ for every $s$, and the setwise top condition gives $H(s,1)=[Q]$ for every $s$, by [L12] and the orbit description [L9]. At $s=0$ and $s=1$, $H$ is respectively the slice of $\beta$ and of ${\beta }^{\mathrm{\prime }}$ by [L3, L13]. The switch $(t,u)\mapsto (u,t)$ is continuous by the product-topology criterion [L7], so $K(t,u):=H(u,t)$ is jointly continuous and is a path homotopy relative to its endpoints by [L6]. Therefore the slice classes agree, and slicing descends to braid-isotopy classes.

Slice after trace is the original loop. For any $[\alpha ]$, the trace lemma says that the unordered slice of the traced braid equals $\alpha (t)$ at every $t$ [L1]. Thus $S(T([\alpha ]))=[\alpha ]$ in the based path-homotopy class set.

Trace after slice is the original braid. By [L11], the ordered coordinate path $t\mapsto (z_1(t),\dots ,z_n(t))$ is a continuous path in $F_n(\mathrm{int}{D}^2)$ starting at $Q$, using [L7, L8, L10]. Its projection is $S(\beta )$ by [L3]. It is therefore an ordered lift of $S(\beta )$ from $Q$; uniqueness in [L1] makes the traced braid exactly $\beta$, so $T(S([\beta ]))=[\beta ]$ as a braid-isotopy class.

The two well-defined assignments satisfy both inverse identities by steps 1.3 and 1.4. Consequently tracing and slicing induce mutually inverse bijections on the stated classes. For $n=0$ both configuration spaces and class sets are singletons; for $n=1$ the configuration spaces identify with the disk and there are no collision conditions, so the same constructions apply [L14].