Curves and geometric intersection numbers on the marked disk

Let $D:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1\},\mathrm{\partial }D:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\},D^{\circ }:=D\setminus \mathrm{\partial }D,$ the closed unit disk with its subspace topology from $\mathbb{C}\cong {\mathbb{R}}^2$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$, 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), and fix once and for all a set $\mathrm{\Delta }\subset D^{\circ }$ of $m+1\ge 2$ marked points with the induced topology. Fix also an orientation of $D$, namely the standard one. Write $G:={\pi }_0\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$ for the boundary-fixed mapping class group of the punctured disk of Boundary-fixed mapping class group of a punctured disk, in its smooth version, whose elements are isotopy classes of diffeomorphisms of $D$ fixing $\mathrm{\partial }D$ pointwise and permuting $\mathrm{\Delta }$ setwise. The cited item defines the homeomorphism version; the smooth configuration-space comparison used here is Khovanov–Seidel, Section 3b, equation (3.1), printed p. 19.

Curves. A curve in $(D,\mathrm{\Delta })$ is a subset $c\subseteq D$ of one of the following two kinds.

1. A simple closed curve $c\subset D^{\circ }\setminus \mathrm{\Delta }$ which is essential, that is, not contractible in $D^{\circ }\setminus \mathrm{\Delta }$; it is the image of an embedding $S^1\hookrightarrow D^{\circ }\setminus \mathrm{\Delta }$ (Smooth embeddings).
2. An arc: the image of an embedding $\gamma \text{\hspace{0.17em}⁣}:[0,1]\to D$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, Smooth embeddings) which is transverse to $\mathrm{\partial }D$, meets the boundary and the marked set exactly in its endpoints, ${\gamma }^{-1}(\mathrm{\partial }D\cup \mathrm{\Delta })=\{0,1\},$ and whose interior lies in $D^{\circ }\setminus \mathrm{\Delta }$. The unoriented image is the curve; a curve may meet $\mathrm{\Delta }$ in one or both of its endpoints, may meet $\mathrm{\partial }D$ in one or both of them, and may have both endpoints in $\mathrm{\Delta }$, both on $\mathrm{\partial }D$, or one of each. Arcs are thus embedded smooth submanifolds with boundary of $D$ (Embedded smooth submanifolds with boundary).

Curves are unoriented: $c$ and its image under any orientation-reversing reparametrization are the same curve. Two curves $c_0,c_1$ are isotopic, written $c_0\simeq c_1$, when one can be deformed into the other by an isotopy in $\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$; endpoints on $\mathrm{\partial }D$ may not move during an isotopy. Isotopy of curves is the orbit relation for the identity component of $\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$. The full mapping class group $G$ acts on these isotopy classes and may carry one class to a different class.

Minimal intersection. Let $c_0,c_1$ be curves. They are in minimal intersection when (i) they intersect transversally, (ii) $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$, and (iii) the following disk formulation of the bigon condition holds: for any two points $z_{-}\mathrm{\ne }{z}_{+}$ of $c_0\cap c_1$ that do not both lie in $\mathrm{\Delta }$, and arcs ${\alpha }_0\subseteq c_0$, ${\alpha }_1\subseteq c_1$ with endpoints $z_{-},z_{+}$ and ${\alpha }_0\cap {\alpha }_1=\{z_{-},z_{+}\}$, the open Jordan disk $K$ enclosed by ${\alpha }_0\cup {\alpha }_1$ contains a marked point. Other portions of the curves may lie in $K$; it need not be a component of $D\setminus (c_0\cup c_1)$. Thus a transverse interior crossing contributing a removable bigon is forbidden, and a pair of arcs with common endpoints in $\mathrm{\Delta }$ is allowed to bound a marked-free disk only when the two arcs share both endpoints.

Existence of minimal representatives. Given curves $c_0,c_1$ with $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$ there is a curve $c_1^{\mathrm{\prime }}\simeq c_1$ in minimal intersection with $c_0$: first perturb $c_1$ into transverse position with distinct endpoint germs, giving finitely many intersections. If the disk condition fails, an innermost marked-free bigon can be removed by pushing $c_1$ across it, with support near the bigon and fixing the marked points and the boundary (the bigon removal of Khovanov–Seidel, Section 3a); as each such move decreases the finite number $\mathrm{\mid }{c}_0\cap c_1\mathrm{\mid }$ of intersection points, the process terminates. In the case $c_0\cap c_1\cap \mathrm{\partial }D\mathrm{\ne }\mathrm{\varnothing }$ one first applies the flow extension below.

Choice scope for geometric intersection numbers. Assume AC (The Axiom of Choice) for the representative-independence, isotopy invariance and auxiliary-choice independence of $I$ supplied by Geometric intersection numbers are isotopy invariants ↗. That supplier retains AC for its topological relative-arc inputs. The definitions of the marked disk, curves, isotopy and minimal intersection, and the finite perturbation and bigon-removal construction above do not invoke that input.

The geometric intersection number. Under this assumption, let $c_0,c_1$ be curves with $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$, and choose a minimal-intersection representative $c_1^{\mathrm{\prime }}\simeq c_1$ of $c_1$. The geometric intersection number is the half-integer $I(c_0,c_1):=\mathrm{\mid }(c_0\cap c_1^{\mathrm{\prime }})\setminus \mathrm{\Delta }\mathrm{\mid }+\frac{1}{2}\mathrm{\mid }{c}_0\cap c_1^{\mathrm{\prime }}\cap \mathrm{\Delta }\mathrm{\mid },I(c_0,c_1)\in \frac{1}{2}\mathbb{Z},$ except in the exceptional case that $c_0,c_1$ are simple closed curves with $c_0\simeq c_1$, where one sets $I(c_0,c_1):=2$. Interior intersection points count once and common marked endpoints count one half, so that in particular $I(b_i,b_i)=1$ for an arc $b_i$ joining two marked points. Isotopic arcs have the same endpoint set, since an isotopy fixes every marked point and every boundary point. The number is independent of the chosen representative $c_1^{\mathrm{\prime }}$: this is proved in Geometric intersection numbers are isotopy invariants ↗, together with the invariance of $I$ under isotopies of both arguments. The value is finite because two curves in minimal intersection meet in finitely many points after a small perturbation, the arcs involved being compact.

The flow extension. The definition above requires $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$. For the source's $I$ on arbitrary pairs one fixes a nonvanishing smooth vector field on $\mathrm{\partial }D$ which is positively oriented with respect to the fixed orientation, extends it to a smooth vector field $Z$ on $D$ which vanishes on $\mathrm{\Delta }$, and lets $(f_t)$ be the flow of $Z$. For $t>0$ small enough that the endpoints of $c_0$ on $\mathrm{\partial }D$ are moved along $\mathrm{\partial }D$ past no endpoint of $c_1$, set $I(c_0,c_1):=I(c_0^{+},c_1),c_0^{+}:=f_t(c_0)\mathrm{.}$ This extension is independent of the auxiliary choices by the isotopy invariance proved in Geometric intersection numbers are isotopy invariants ↗; it depends on the orientation of $D$ and is not symmetric, since it removes the common boundary endpoints of $c_0$ with $c_1$ by pushing $c_0$ off them.

Standing conventions. Under the stated AC hypothesis, throughout this page $I$ always denotes this half-integer valued function of isotopy classes of curves, with the exceptional value $2$ for isotopic simple closed curves, and with the flow extension whenever a pair meets on $\mathrm{\partial }D$. The basic arcs $b_0,\dots ,b_m$ fixed in Basic arcs, admissible curves and the standard normal form form a chain: $b_0$ joins a fixed boundary point to the first marked point, and $b_i$ for $1\le i\le m$ joins the $(i-1)$st marked point to the $i$th marked point, and all values $I(b_j,c)$ quoted on this page are computed in that fixed picture, whose data are fixed there once and for all.