Standard meridians of a punctured disk

Let $n\in \mathbb{N}$, let $D^2=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1\}$ and $Q_n=(q_1,\dots ,q_n)$ be the closed unit disk and the base configuration of Boundary-fixed mapping class group of a punctured disk, so that $q_1<\cdots <q_n$ are distinct points of the real axis of $\mathrm{int}{D}^2$. Fix the boundary basepoint $d:=(0,1)\in \mathrm{\partial }{D}^2\mathrm{.}$ For $1\le i\le n$ let $s_i\text{\hspace{0.17em}⁣}:[0,1]\to D^2$ be the straight stem $s_i(t):=(1-t)d+t{q}_i,$ the straight segment from $d$ to $q_i$, and choose a round circle $C_i$ in $D^2\setminus Q_n$ centred at $q_i$, of radius $0<{\epsilon }_i<d(q_i,\mathrm{\partial }{D}^2)$ so small that the closed disks $B_i$ bounded by the circles are pairwise disjoint, that each circle meets only its own stem, and that it meets $s_i$ exactly once, at $p_i:=s_i(1-{\epsilon }_i\mathrm{/}{L}_i)$ where $L_i:=\mathrm{\mid }d-q_i\mathrm{\mid }$, and meets no other stem $s_j$ with $j\mathrm{\ne }i$. Write $c_i\text{\hspace{0.17em}⁣}:[0,1]\to D^2\setminus Q_n$ for $C_i$ traversed once positively (counterclockwise), from $p_i$ back to $p_i$. The standard meridian loops are the loops based at $d$ $x_i:=s_i{c}_i{s}_i^{-1},1\le i\le n,$ read as $s_i$ from $d$ to $p_i$, then $c_i$, then $s_i$ from $p_i$ back to $d$ (Based loops and the fundamental group). Further, $\mathrm{\partial }$ denotes the positively oriented boundary loop $\mathrm{\partial }(t):=d{e}^{2\pi it},t\in [0,1],$ of $D^2$ based at $d$: it traverses $\mathrm{\partial }{D}^2$ once counterclockwise, starting and ending at $d$.

Existence and independence of the choices. The straight segments $s_i$ leave $d$ in pairwise distinct directions (the points $q_1,\dots ,q_n$ are distinct and lie strictly below $d$) and meet one another only at $d$; the segments $s_i$ meet the real axis only at their endpoints $q_i$. An explicit choice-free family is given by ${\epsilon }_i=\frac{1}{4}\min (\{1-\mathrm{\mid }{q}_i\mathrm{\mid }\}\cup \{\mathrm{\mid }{q}_i-q_j\mathrm{\mid },\frac{\mathrm{\mid }{q}_i-q_j\mathrm{\mid }}{\sqrt{1+q_j^2}}:j\mathrm{\ne }i\})\mathrm{.}$ The finite set inside the minimum is nonempty and contains only positive numbers. The distance from $q_i$ to the line through $d,q_j$ is $\mathrm{\mid }{q}_i-q_j\mathrm{\mid }\mathrm{/}\sqrt{1+q_j^2}$, so $B_i$ misses every other stem. Also ${\epsilon }_i+{\epsilon }_j\le \mathrm{\mid }{q}_i-q_j\mathrm{\mid }\mathrm{/}2<\mathrm{\mid }{q}_i-q_j\mathrm{\mid }$, proving disjointness of the closed disks, and ${\epsilon }_i<1-\mathrm{\mid }{q}_i\mathrm{\mid }$ keeps each disk inside $D^2$. The radius is less than $\mathrm{\mid }d-q_i\mathrm{\mid }$, giving exactly the displayed contact with its own stem. For $n=1$ the minimum has only its boundary-margin entry; for $n=0$ the family is empty. Any family satisfying these conditions may be fixed; this formula witnesses its existence without any choice axiom.

The class $[x_i]\in {\pi }_1(D^2\setminus Q_n,d)$ is independent of all admissible radii, including the old broader convention requiring only avoidance of the other stems. Such a circle encloses no $q_j$ with $j\mathrm{\ne }i$: otherwise the other stem from $d$, which lies outside the circle, to $q_j$ would cross it (or have $q_j$ on it). Thus the larger disk of any two admissible concentric circles still contains no other puncture. Interpolate their radii and their tether contact points radially; this is a based lasso homotopy in $D^2\setminus Q_n$ (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints). It does not require the intermediate circle to be disjoint from the other circles, so the class argument does not circularly assume the newly explicit representative convention. Fix once and for all one such family of circles; every statement on this page uses that fixed family.

Frozen conventions of the page. The stems and the basepoint transport are fixed once and for all by the choices above; the stems are indexed so that the positively oriented boundary loop $\mathrm{\partial }$, as seen from $d$, meets the directions of the stems in the order $s_1,s_2,\dots ,s_n$. Products of braid automorphisms use ordinary function composition: the leftmost factor is the outermost map, so the rightmost factor acts first. No relation of the braid presentation and no choice principle is used in this definition.

• The index convention is a property of the labelling of the punctures: the directions of the stems from $d$ occur in the same cyclic order as the punctures on the real axis, so that tracing the counterclockwise boundary from $d$ meets the angular positions of $s_1,\dots ,s_n$ in increasing index order.
• The loops $x_1,\dots ,x_n$ are the loops denoted $x_1,\dots ,x_n$ in Figure 3 of Boundary-fixed mapping class group of a punctured disk's source (Gonzalez-Meneses, section 1.6) and correspond to Artin's generators $t_1,\dots ,t_n$ of the free group of the punctured disk.