18 Figure-eight curves and simple lifts

18.1 The pants trace relation

Let $\gamma$ be a closed geodesic with exactly one transverse self-intersection, not a proper iterate. Its two-lobed regular neighborhood is a pair of pants: the two ends of each lobe are adjacent in the cyclic order at the crossing. Its cuffs are essential or peripheral; a compressing disk would give a geodesic monogon or bigon, or make the curve an iterate. After tightening the cuffs, allow coincident cuff images in the ambient surface; this occurs on a one-holed torus. If the three cuff lengths are $a,b,c$, write $$t_a=2\cosh(a/2),\quad t_b=2\cosh(b/2),\quad t_c=2\cosh(c/2).$$ A cusp has positive geometric trace $2$, regardless of the sign of a chosen $\mathrm{SL}_2$ lift. For the figure-eight that wraps the $a$ and $b$ cuffs, the trace identity is $$\boxed{t_\gamma=t_at_b+t_c.}$$(18.1) This is Rivin’s pants relation [90, Theorem 1.3]. It also follows directly from $\mathop{\mathrm{tr}}(AB^{-1})=\mathop{\mathrm{tr}}A\,\mathop{\mathrm{tr}}B-\mathop{\mathrm{tr}}(AB)$ with the consistent signs of pants holonomies: choose $\mathop{\mathrm{tr}}A=t_a$, $\mathop{\mathrm{tr}}B=t_b$, and $\mathop{\mathrm{tr}}(AB)=-t_c$. Rewriting in positive geometric traces produces the plus sign in (18.1). One cannot make this replacement in an arbitrary character identity; Section 23 retains signed traces and therefore has a different sign.

For pants with lengths $(a,b,0)$, the two figures that use the cusp as one lobe have traces $$u=2t_a+t_b,\qquad v=t_a+2t_b, \qquad t_a=\frac{2u-v}{3},\quad t_b=\frac{2v-u}{3}.$$(18.2) Consequently any weight assigned to this pair of cuffs can be rewritten as a weight in the two associated figure-eight lengths. This gives a direct reindexing of a pants identity. It produces a paired-curve weight; it does not automatically produce a universal function of one figure-eight length.

18.2 What the local double cover accomplishes

Use pants generators $a,b$, with third cuff $c=(ab)^{-1}$ and figure $\gamma=ab^{-1}$. Consider the connected double cover given by $$\chi(a)=\chi(b)=1\in\mathbb Z/2\mathbb Z.$$ Then $\chi(c)=\chi(\gamma)=0$. The $a,b$ cuffs each lift with degree two, while $c$ has two degree-one lifts. The upstairs surface is a four-holed sphere with cuff lengths $$(2a,2b,c,c).$$(18.3) The figure-eight has two closed degree-one lifts, each of which is simple. One way to see this is to lift its two-loop spine: the lifted graph has two vertices, and either lift uses one $a$ edge and one $b$ edge to form a simple cycle.

The two lifted cycles meet at both vertices of that graph. Geometrically, the two simple lifts intersect each other twice, once over each lift of the original crossing. Thus two components of the lifted domain do not mean a disjoint multicurve upstairs.

There is a further global issue: this particular character of the pants group need not extend to the whole surface. A lobe that is zero in $H_1(S;\mathbb Z/2\mathbb Z)$—for example, a separating lobe on a closed surface— cannot be assigned value one by an ambient character. Rivin proves a global substitute: a one-double-point geodesic has a simple lift in a fourfold cover [90, Theorem 4.2].

Even when a suitable cover exists, applying an entire simple-curve identity upstairs and projecting downstairs introduces other projected topological types, covering degrees, and multiplicities. Desingularization supplies an important geometric tool, but selecting just the desired downstairs curves requires additional accounting.

18.3 The two types on a punctured torus

On a once-punctured torus, a figure-eight has a unique essential simple curve $\alpha$ disjoint from it. To see the topological restriction, truncate the cusp and consider the complement of its incompressible pants neighborhood in the resulting compact torus. That complement has Euler characteristic zero and no disk components, so it consists of two annuli. One joins a pants cuff to the cusp boundary; the other joins the remaining two cuffs, which are therefore parallel essential curves. Conversely, any essential simple curve disjoint from the figure can be pushed out of its regular neighborhood and is parallel to these cuffs. Thus its pants neighborhood has one peripheral cuff and two essential cuffs that tighten to $\alpha$. Cutting along $\alpha$ gives pants of lengths $(a,a,0)$, where $a=\ell_\alpha$ and $t=2\cosh(a/2)>2$.

There are exactly three unoriented figure-eight classes in these cut pants. Indeed, splitting at the unique crossing gives two simple based lobes; their two distinct peripheral classes determine a choice of two of the three pants cuffs. Traversing them with opposite boundary orientations is the one-crossing class; traversing their product with the other relative orientation is the third cuff. This gives the three choices below:

The first two are exchanged by the hyperelliptic involution; call their family $\mathcal N$. The third is hyperelliptic; call its family $\mathcal H$. Equal lengths in the first row do not identify the two free-homotopy classes. In torus homology the first family has a primitive class up to sign and the second twice a primitive class, although its geodesics are not proper iterates.

For a word check, glue pants generators by $b=s a^{-1}s^{-1}$. The first two figures can be represented by $a^2b$ and $ab^2$, while the third is $ab^{-1}=asa s^{-1}$. The last word is not a square. The common disjoint class $\alpha$ is recoverable from the figure’s pants neighborhood, so grouping all figure-eights by $\alpha$ has no duplication. More explicitly, the first two words become $a^2sa^{-1}s^{-1}$ and $asa^{-2}s^{-1}$ in the free basis $a,s$. Their cyclic words are neither conjugate nor inverse-conjugate. Their homology classes are $[a]$ and $-[a]$; the third has class $2[a]$ and cyclic word $asa s^{-1}$, which is not a power of a shorter cyclic word. This also checks primitivity.

Lei–Zhang correct the index set in Rivin’s earlier torus identity: its stated formula accounts for $\mathcal N$, while $\mathcal H$ requires separate treatment [69, p. 2350049-1, abstract]. The classification and weights used here have been derived above; this citation does not substitute for a check of the full correction paper. The separation is important here because the next construction is intended to include both families.

18.4 A positive single-length kernel for all figure-eights

Define, for $t>2$, $$h(t)=1-\sqrt{1-\frac4{t^2}}.$$(18.4) If $t=2\cosh(\ell/2)$, then $h(t)=2/(1+e^\ell)$. Hence McShane’s identity is $\sum_\alpha h(t_\alpha)=1$. The classification immediately gives the two separate identities $$\begin{aligned} \sum_{\gamma\in\mathcal N} \left(1-\sqrt{1-\frac{36}{t_\gamma^2}}\right)&=2, \\ \sum_{\gamma\in\mathcal H} \left(1-\sqrt{1-\frac4{t_\gamma-2}}\right)&=1. \end{aligned}$$

To use the same kernel on every one-crossing geodesic, let $$\beta(x)=\frac{x^2}{9}+2,\qquad F(x)=\sum_{n=0}^{\infty}(-1)^n2^{-n-1} h\left(\frac{\beta^{\circ n}(x)}3\right),\qquad x>6.$$(18.7) Here $\beta^{\circ n}$ means iteration, with $\beta^{\circ0}(x)=x$.

The same alternating-series argument supplies an explicit finite evaluation contract. For $N\geq1$ let $F_N$ be the sum of the first $N$ terms in (18.7), and put $x_N=\beta^{\circ N}(x)$. Then $$0\leq(-1)^N\bigl(F(x)-F_N(x)\bigr) \leq 2^{-N-1}h(x_N/3)\leq 2^{-N-1}.$$(18.11) Moreover the exact finite identity $$2F_N(x)+F_N(\beta(x))-h(x/3) =(-1)^{N-1}2^{-N}h(x_N/3)$$(18.12) provides a check independent of extrapolating a computed infinite sum. For large $t$, evaluate $h(t)=4/[t^2(1+\sqrt{1-4/t^2})]$ to avoid subtraction of nearby numbers. These bounds concern evaluation of the kernel. Approximating the sum over surface curves additionally requires a complete index enumeration and a bound for the omitted curve classes; no such numerical approximation is asserted here.

This is an elementary construction from the corrected classification, not a claim of a new identity in the literature. It is restricted to the cusped torus. Cone continuation requires checking existence of the indexed geodesics as well as continuing the weight; the cone obstruction discussed by Lei–Zhang is a reason to keep that distinction explicit [69].