Basic arcs, admissible curves and the standard normal form

Let $(D,\mathrm{\Delta })$ be the marked disk of Curves and geometric intersection numbers on the marked disk with $m+1\ge 2$ marked points, and let $G={\pi }_0\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$ be its boundary-fixed mapping class group (Boundary-fixed mapping class group of a punctured disk). Write $\mathcal{D}:=\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$ for the actual diffeomorphism group, so $G={\pi }_0(\mathcal{D})$. Actual curve images below use members of $\mathcal{D}$, and mapping classes act on curve-isotopy classes. All curves below are curves in $(D,\mathrm{\Delta })$ in the sense of that definition.

Basic sets. Fix the boundary endpoint $d$ of the standard drawn chain of source Figure 2, and label its marked points $q_0,\dots ,q_m$ along that chain. The standard basic arcs are $b_0$ from $d$ to $q_0$ and $b_i$ from $q_{i-1}$ to $q_i$ for $1\le i\le m$. Their interiors are pairwise disjoint and avoid $\mathrm{\Delta }\cup \mathrm{\partial }D$; consecutive arcs meet only at their common marked endpoint, and all other arcs are disjoint. A basic set of curves is an actual image of this fixed chain under a member of $\mathcal{D}$. The marked labels are transported with it. Under AC, the preferred identification of the Artin-presentation group $B_{m+1}$ with $G$ sends ${\sigma }_i$ to the half twist about $b_i$, $1\le i\le m$. Presentation completeness is supplied by The Artin presentation is complete for geometric braids; the geometric-to-topological comparison is Braid group as boundary-fixed punctured-disk mapping classes. For the smooth group $G$ used here, use the smooth configuration-space comparison and generator convention of Khovanov–Seidel, Section 3b, equation (3.1) and Figure 6, printed pp. 19–20 (The Axiom of Choice, The elementary geometric half twist, its support disc, and its opposite). Admissible curves. A curve $c$ is admissible if $c=f(b_i)$ for some $f\in \mathcal{D}$ and some $i$, that is, if it lies in the $\mathcal{D}$-orbit of the actual basic set. The endpoints of an admissible curve lie in $\mathrm{\Delta }\cup (b_0\cap \mathrm{\partial }D)$, and conversely every arc with such endpoints is admissible; $G$ acts on the set of isotopy classes of admissible curves.

Vertical curves and normal form. Fix arcs $d_0,\dots ,d_m$ as in Figure 11, pairwise disjoint embedded arcs dividing $D$ into regions $D_0,D_1,\dots ,D_{m+1}$ in that order, chosen so that $d_k$ meets the spine in the prescribed way of the standard picture. An admissible curve $c$ is in normal form if it has minimal intersection with every $d_k$. Every admissible curve can be isotoped into normal form, and the normal form is unique up to isotopy preserving each $d_k$ setwise and fixing the marked points and $\mathrm{\partial }D$ (the source's Lemma 3.15); consequently the combinatorial data below are invariants of the isotopy class of $c$.

Crossings, segments, strings. For an admissible curve $c$ in normal form put $\mathrm{cr}(c):=c\cap (d_0\cup \cdots \cup d_m),$ the set of crossings of $c$; a crossing lying on $d_k$ is a $k$-crossing. The connected components of $c\cap D_k$ are the segments of $c$; a segment is essential when both its endpoints are crossings, and inessential otherwise (it then ends at a point of $\mathrm{\Delta }\cup \mathrm{\partial }D$). The connected components of $c\cap (D_k\cup D_{k+1})$, for $0\le k\le m$ with $D_{m+1}$ interpreted as the region across $d_m$ in the standard picture, are the $k$-strings of $c$; write $\mathrm{st}(c,k)$ for their set. By the uniqueness of normal form, the number and relative position of the segments and of the strings are invariants of $c$.

String types. Up to isotopy of $D_k\cup D_{k+1}$ fixing the boundary arcs and the marked points, each $k$-string belongs to one of the following families or exceptional types. For $1\le k<m$ there are five infinite families ${\mathrm{I}}_u,{\mathrm{I}\mathrm{I}}_u,{\mathrm{I}\mathrm{I}}_u^{\mathrm{\prime }},{\mathrm{I}\mathrm{I}\mathrm{I}}_u,{\mathrm{I}\mathrm{I}\mathrm{I}}_u^{\mathrm{\prime }}(u\in \mathbb{Z})$ and five exceptional types $\mathrm{I}\mathrm{V},{\mathrm{I}\mathrm{V}}^{\mathrm{\prime }},\mathrm{V},{\mathrm{V}}^{\mathrm{\prime }},\mathrm{V}\mathrm{I}$ (Figures 15-16); the type ${\mathrm{I}}_{u+1}$ is obtained from ${\mathrm{I}}_u$ by applying the half twist about $b_k$, and likewise for the other families. For $k=m$ the list consists of the two families ${\mathrm{I}\mathrm{I}}_u,{\mathrm{I}\mathrm{I}\mathrm{I}}_u$ and the two exceptional types $\mathrm{V},\mathrm{V}\mathrm{I}$ (Figure 17), and for $k=0$ of the five exceptional types $\mathrm{V}\mathrm{I}\mathrm{I},\mathrm{V}\mathrm{I}\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{X},\mathrm{X},\mathrm{X}\mathrm{I}$ (Figure 18). The members $u=0$ of the families and the exceptional types are the drawn models of those figures; each type is an isotopy class of arcs in $D_k\cup D_{k+1}$ with the prescribed endpoints on $d_{k-1},d_k,d_{k+1}$ and $\mathrm{\Delta }\cup \mathrm{\partial }D$.

Segments types. A segment of $c\cap D_k$ for $1\le k\le m$ is, up to the analogous isotopy, of one of the six types labelled $1,1^{\mathrm{\prime }},2,2^{\mathrm{\prime }},3,3^{\mathrm{\prime }}$ of Figure 12; for $k=m+1$ there are the two types analogous to $2$ and $3$ (Figure 13), and for $k=0$ the single type of Figure 14. The essential segments are precisely those of type $1,1^{\mathrm{\prime }},2,2^{\mathrm{\prime }}$; the basic curves $b_0,\dots ,b_m$ themselves have no essential segments.

The nested twists. Fix the pairwise disjoint nested curves $l_0,\dots ,l_{m-1}$ of Figure 7. The disk bounded by $l_j$ contains exactly the suffix of marked points $\{q_j,\dots ,q_m\}$. Choose a small closed annular neighbourhood of $l_j$, disjoint from all marks and from the other support annuli; among the basic arcs it meets only $b_j$, in one transverse crossing. Let ${\tau }_j$ be a positive Dehn twist supported in that annulus. Its class lies in $G$, the classes commute by disjoint support, and the chosen representative fixes every $b_k$ with $k\mathrm{\ne }j$. They generate the standard twist subgroup used in the detector lemma below. Standing convention. All the data above — the basic set $b_0,\dots ,b_m$, the vertical curves $d_0,\dots ,d_m$, the regions $D_0,\dots ,D_{m+1}$ and the nested curves $l_0,\dots ,l_{m-1}$ — are fixed once and for all in the standard picture, and every statement invoking them names this fixed picture. When a statement is applied to an arbitrary basic set, it is transported by an actual diffeomorphism in $\mathcal{D}$ carrying the standard basic set to the given one; the transport is part of the statement.