Braid isotopy relative to the top and bottom endpoints

Let $n\in \mathbb{N}$ and let $Q=(q_1,\dots ,q_n)$ be the base configuration of Geometric braids in the disc with setwise endpoints, so that braids based at $Q$ are tuples of continuous maps $z_j\text{\hspace{0.17em}⁣}:I\to D^{\circ }$ satisfying conditions (1)--(3) of that definition. Write $I=[0,1]$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and give $I\times I$ the product topology (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

Let $\beta =(z_1,\dots ,z_n)$ and ${\beta }^{\mathrm{\prime }}=(z_1^{\mathrm{\prime }},\dots ,z_n^{\mathrm{\prime }})$ be braids based at $Q$. A braid isotopy from $\beta$ to ${\beta }^{\mathrm{\prime }}$ relative to the top and bottom endpoints, or simply a braid isotopy, is an $n$-tuple

$Z=(Z_1,\dots ,Z_n),Z_j\text{\hspace{0.17em}⁣}:I\times I⟶D^{\circ },$

of continuous maps (Continuity of a map of topological spaces at a point and globally) such that

1. for every $s\in I$, the tuple $Z(s,\cdot ):=(Z_1(s,\cdot ),\dots ,Z_n(s,\cdot ))$ is a braid based at $Q$, i.e. $t\mapsto Z_j(s,t)$ is continuous into $D^{\circ }$, the $n$ values $Z_j(s,t)$ are pairwise distinct for every $t$, the bottom condition $Z_j(s,0)=q_j$ holds for every $s$ and $j$, and $\{Z_1(s,1),\dots ,Z_n(s,1)\}=\{q_1,\dots ,q_n\}$ for every $s$;
2. $Z_j(0,t)=z_j(t)$ and $Z_j(1,t)=z_j^{\mathrm{\prime }}(t)$ for all $j$ and $t$.

The first variable $s$ is the isotopy parameter and the second variable $t$ is the height; a family of motions depending on $s$ is a braid isotopy exactly when the map $(s,t)\mapsto Z_j(s,t)$ is jointly continuous, not merely continuous in $s$ for each fixed $t$. When such a $Z$ exists we say that $\beta$ and ${\beta }^{\mathrm{\prime }}$ are braid-isotopic and write $\beta \sim {\beta }^{\mathrm{\prime }}$.

Relation to homotopy. A braid is a continuous map $I\to (D^{\circ }{)}^n$ with additional properties, and a braid isotopy is a homotopy of such maps (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints) that is relative to the bottom subset $\{0\}\subseteq I$, in the following sense: the homotopy condition $Z_j(s,0)=z_j(0)=z_j^{\mathrm{\prime }}(0)$ for all $s$ says that the whole bottom point is fixed throughout the deformation. At the top, the condition is imposed setwise: $\{Z_1(s,1),\dots ,Z_n(s,1)\}=\{q_1,\dots ,q_n\}$. Joint continuity then forces each individual top endpoint to remain constant as $s$ varies, since a continuous map from an interval into this finite discrete set is constant.

Endpoint permutations of the slices. Let $Z$ be a braid isotopy from $\beta$ to ${\beta }^{\mathrm{\prime }}$. Each slice $Z(s,\cdot )$ is a braid and hence has an endpoint permutation $\pi (Z(s,\cdot ))\in S_n$ (Geometric braids in the disc with setwise endpoints, The finite symmetric group $S_n$, one-line notation, and cycle notation). This definition does not separately impose that these slice permutations agree: it requires each slice to be a braid and the family to be jointly continuous. The slice permutations do agree, and therefore the endpoint permutation is an invariant of braid isotopy, with $\pi (\beta )=\pi ({\beta }^{\mathrm{\prime }})$; this is proved on this page, in the stacking proposition, by a connectedness argument for the height interval $I$.

Isotopy of the ambient cylinder is not enough. The definition constrains the deformation to the product $D^{\circ }\times I$ through height-preserving motions; it is neither an isotopy of an arbitrary embedded link nor a free homotopy of the tuple of paths. Deformations that make a strand meet a horizontal plane more than once, or that move bottom points, are not braid isotopies; a counterexample is recorded on the companion examples page.