Geometric Braids and Artin Generators

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.

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.

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, with $h:=\frac{1}{4(n+1)}$ and $q_j=((2j-n-1)h,0)$. Let $\beta =(z_1,\dots ,z_n)$ and $\gamma =(w_1,\dots ,w_n)$ be braids based at $Q$ with endpoint permutations $\pi (\beta )$ and $\pi (\gamma )$ (Geometric braids in the disc with setwise endpoints), and let $\sim$ denote braid isotopy relative to the top and bottom (Braid isotopy relative to the top and bottom endpoints).

(a) The stacked tuple is a braid. Define $\gamma \star \beta$ by

$(\gamma \star \beta {)}_j(t):=\{\begin{array}{ll}{z}_j(2t),& 0\le t\le \frac{1}{2},\\ w_{\pi (\beta )(j)}(2t-1),& \frac{1}{2}\le t\le 1.\end{array}$

Then $\gamma \star \beta$ is a braid based at $Q$. It is the stacking of $\beta$ below $\gamma$: the motion $\beta$ runs during the first half of the height interval and the motion $\gamma$ during the second half, after the strand of label $\pi (\beta )(j)$ has been joined.

(b) The endpoint permutation multiplies. $\pi (\gamma \star \beta )=\pi (\gamma )\circ \pi (\beta )$, with $(\pi \circ \sigma )(j)=\pi (\sigma (j))$.

(c) The operation descends to isotopy classes. If $\beta \sim {\beta }^{\mathrm{\prime }}$ and $\gamma \sim {\gamma }^{\mathrm{\prime }}$ then $\gamma \star \beta \sim {\gamma }^{\mathrm{\prime }}\star {\beta }^{\mathrm{\prime }}$.

(d) The operation is associative. If $\delta$ is a further braid based at $Q$, then $(\delta \star \gamma )\star \beta \sim \delta \star (\gamma \star \beta )$.

(e) The endpoint permutation is constant along isotopies. If $\beta \sim {\beta }^{\mathrm{\prime }}$ then $\pi (\beta )=\pi ({\beta }^{\mathrm{\prime }})$; equivalently $\pi$ is constant on each braid isotopy class, so that $\pi$ is a function of the class $[\beta ]$ alone.

Consequently the stacking of isotopy classes, $[\gamma ][\beta ]:=[\gamma \star \beta ]$, is a well-defined associative binary operation on the set of braid isotopy classes based at $Q$, with $\pi (\delta \star \gamma \star \beta )=\pi (\delta )\pi (\gamma )\pi (\beta )$ for any three braids.

• The reparametrisation used in step 3.1 is the only place where associativity needs work: the two bracketings of a threefold stack differ by how the height interval is cut, and a continuous family of cuts is an isotopy because the underlying sequence of motions, and all label couplings, are unchanged.
• Clause (e) is what makes the endpoint permutation a function of the isotopy class rather than of the representative; it is the connectedness of the height interval, through [L4], that rules out a strand ending at a different base point at the two ends of an isotopy.
• Nothing in the proposition uses a choice principle: the base configuration, and every reparametrisation, is given by an explicit formula, and the two-element cover of $I$ in step 1.1 is finite.

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. Write

$G_n:=\{[\beta ]:\beta \text{ a braid based at }Q\}$

for the set of braid isotopy classes relative to the top and bottom (Braid isotopy relative to the top and bottom endpoints), and let $[\gamma ][\beta ]:=[\gamma \star \beta ]$ be the stacking of Stacking of geometric braids is a well-defined associative operation on isotopy classes. Then:

(a) $G_n$ is a group (Group and abelian group) with this operation. Its identity is the class $[e]$ of the trivial braid $e_j(t):=q_j$, and the inverse of $[\beta ]$, for $\beta =(z_1,\dots ,z_n)$ with endpoint permutation $\pi (\beta )$, is the class of the reversed braid

${\bar{\beta }}_j(t):=z_{\pi (\beta {)}^{-1}(j)}(1-t)(t\in I,1\le j\le n)\mathrm{.}$

(b) The endpoint permutation map

$\pi \text{\hspace{0.17em}⁣}:G_n⟶S_n,[\beta ]⟼\pi (\beta ),$

is a well-defined group homomorphism (The finite symmetric group $S_n$, one-line notation, and cycle notation).

The construction is choice-free: all motions and reparametrisations used are given by explicit formulas.