Geometric braids in the disc with setwise endpoints

Throughout this page $n\in \mathbb{N}$ is a natural number (The natural numbers $\mathbb{N}$ (von Neumann)) with the labels $1,\dots ,n$, and

$I:=[0,1]\subseteq \mathbb{R}$

is the unit interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). Points of ${\mathbb{R}}^2$ are written as vectors and are added and scaled coordinatewise. The closed unit disc is

$D:={\bar{B}}_2(0,1)=\{w\in {\mathbb{R}}^2:\parallel w{\parallel }_2\le 1\}$

and its interior is $D^{\circ }:=\{w\in {\mathbb{R}}^2:\parallel w{\parallel }_2<1\}$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$), with the subspace topology inherited from ${\mathbb{R}}^2$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace); the cylinder $D^{\circ }\times I$ carries 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) and its subspace topology. Continuous means continuous with respect to these topologies (Continuity of a map of topological spaces at a point and globally).

The base configuration. Put

$h:=\frac{1}{4(n+1)},q_j:=((2j-n-1)h,0)\in {\mathbb{R}}^2(1\le j\le n),$

and let $Q:=(q_1,\dots ,q_n)$. The points $q_1,\dots ,q_n$ lie in $D^{\circ }$, listed strictly left to right, and are equally spaced:

$\parallel q_j{\parallel }_2=(n+1-2j)h\text{ for }j\le \frac{n+1}{2},\parallel q_j{\parallel }_2=(2j-n-1)h\text{ for }j\ge \frac{n+1}{2},$

so that $\parallel q_j{\parallel }_2\le (n-1)h<\frac{1}{4}$ and $q_{j+1}-q_j=(2h,0)$ for every $j$. In particular $q_i\mathrm{\ne }{q}_j$ for $i\mathrm{\ne }j$, and $Q$ is an ordered tuple of pairwise distinct points of $D^{\circ }$. The configuration $Q$ is fixed once and for all on this page and is not part of the data of a braid.

The label set. The labels $1,\dots ,n$ are part of the data. Throughout this page and its companion they are identified with the set $n=\{0,1,\dots ,n-1\}$ of predecessors of $n$ (The natural numbers $\mathbb{N}$ (von Neumann)) by the bijection $\kappa (i):=i-1$, and it is through $\kappa$ that the symmetric group $S_n=\mathrm{Sym}(n)$ of The finite symmetric group $S_n$, one-line notation, and cycle notation acts on the labels, with composition read with the right-hand factor first. Thus a symbol such as $(ii+1)$ denotes the transposition exchanging the labels $i$ and $i+1$ for $1\le i\le n-1$, the symbol $\mathrm{id}$ denotes the identity permutation of the labels, and a bijection of $\{1,\dots ,n\}$ is regarded as an element of $S_n$ through $\kappa$.

Geometric braid. A geometric braid on $n$ strands based at $Q$, or simply a braid, is an $n$-tuple

$\beta =(z_1,\dots ,z_n)$

of continuous maps $z_j\text{\hspace{0.17em}⁣}:I\to D^{\circ }$ (Continuity of a map of topological spaces at a point and globally) such that

1. $z_i(t)\mathrm{\ne }{z}_j(t)$ whenever $i\mathrm{\ne }j$ and $t\in I$;
2. $z_j(0)=q_j$ for every $j$;
3. $\{z_1(1),\dots ,z_n(1)\}=\{q_1,\dots ,q_n\}$.

The $j$-th strand of $\beta$ is the graph $\{(z_j(t),t):t\in I\}\subseteq D^{\circ }\times I$, and the second coordinate $t$ is its height. The defining conditions say that each strand meets every horizontal slice ${\mathbb{R}}^2\times \{t\}$ in exactly one point, that no two strands meet, that the bottom endpoints are the labelled base points $q_1,\dots ,q_n$, and that the top endpoints form the base configuration setwise. A braid is called pure when in addition $z_j(1)=q_j$ for every $j$.

This parametrised, level-preserving presentation is the object used in this page: the $z_j$ are the point motions, and the strands are recovered as their graphs. It is not an unqualified tame link in the cylinder. The moving points stay in the interior $D^{\circ }$ of the disc, and the base configuration is chosen in $D^{\circ }$: this interior convention gives every motion a positive distance from the boundary circle $\mathrm{\partial }D=S^1$, which the later polygonal approximation uses.

Endpoint permutation. Let $\beta =(z_1,\dots ,z_n)$ be a braid. For each $j$ the setwise condition (3) produces at least one index $\pi (j)$ with $z_j(1)=q_{\pi (j)}$, and the pairwise distinctness of $q_1,\dots ,q_n$ makes it unique; moreover $j\mapsto \pi (j)$ is injective, because $q_{\pi (j)}=z_j(1)$ shows that distinct $j$ give distinct points $q_{\pi (j)}$. Hence $j\mapsto \pi (j)$ is a bijection of $\{1,\dots ,n\}$, that is, a permutation (The finite symmetric group $S_n$, one-line notation, and cycle notation). It is called the endpoint permutation of $\beta$ and is written $\pi (\beta )\in S_n$:

$z_j(1)=q_{\pi (\beta )(j)}(1\le j\le n)\mathrm{.}$

Labels are transported from the bottom: the $j$-th strand is the one that starts at $q_j$, and $\pi (\beta )(j)$ records where it ends. Thus $\beta$ is pure exactly when $\pi (\beta )=\mathrm{id}$, while the setwise condition (3) alone allows $\pi (\beta )\mathrm{\ne }\mathrm{id}$. The adjective setwise in the title refers to condition (3); it is not a purity assumption.

Elementary cases and the trivial braid. For $n=0$ the tuple is empty, the conditions are vacuous, and there is exactly one braid, the empty tuple; its endpoint permutation is the unique element of $S_0$. For $n=1$ we have $q_1=(0,0)$, and a braid is exactly a continuous path $z_1\text{\hspace{0.17em}⁣}:I\to D^{\circ }$ with $z_1(0)=z_1(1)=(0,0)$, the endpoint permutation being the identity of $S_1$. For every $n$ the trivial braid $e$ is the tuple of constant motions $z_j(t):=q_j$; it is pure, since $e_j(1)=q_j=q_{\mathrm{id}(j)}$ for every $j$.

Slicing is continuous by construction. Because each $z_j$ is a genuine function of the height with $z_j(0)=q_j$, the bottom endpoints are fixed pointwise, and the top condition is imposed only on the set of top endpoints. This is the distinction used by Braid isotopy relative to the top and bottom endpoints: an isotopy of braids must keep each bottom point fixed and the top configuration setwise equal to $Q$, but it need not return each strand to its own starting point.