The object of size 24

The object of size 24, stabilizer $C_7$ , one class of 8 subgroups · 3 automorphisms

Stabilizer $C_7$ and automorphism group $C_3$ : the flexes of the Klein quartic, where seam monodromy first appears.

The object of size 24 as the 24 flexes of the Klein quartic, in eight flex triangles, each turned by the flex-tangent map

\tau

, an automorphism of power 4.

Its incarnations

The seam table names the object in each of the five theories that meet the group. Every entry there is built.

Fano plane: cyclic labellings mod translation, for $\{0,1,3\}$ and for $\{0,4,6\}$
Projective line: nonzero vectors of $\F_7^2$ up to sign
The group: elements of order 7 (two classes, $7A$ and $7B$ )
Klein quartic: flexes; flex tangents
Graphs: Heawood perfect matchings; Coxeter heptagons

The seams between two incarnations form a torsor under $N_G(H)/H$ , a group of order 3, so there are 3 of them.

The fifteen objects

The object on the line of sizes. In ink, the objects it maps onto; in blue, those that map onto it. Dots count each object’s automorphisms, and dashed brackets join the pairs with one permutation character.

Proposition

$N_G(C_7)=7{:}3$ , so the automorphism group is $C_3$ and there are three seams between any two incarnations. The power applies, with $K_{C_7}=\{1,2,4\}$ , the squares modulo 7: two automorphisms of two incarnations correspond under every seam exactly when their powers agree. Both nontrivial automorphisms have quotient class $7{:}3$ .

Example

The classes $7A$ , of $z\mapsto z+1$ , and $7B$ , of its inverse; the nonzero vectors of $\F_7^2$ up to sign; the flexes and the flex tangents of the Klein quartic; the cyclic labellings of the Fano plane modulo translation, for $\{0,1,3\}$ and for $\{0,4,6\}$ ; the perfect matchings and the 14-cycles of the Heawood graph; the heptagons of the Coxeter graph; the faces of Klein’s map; and the cusps of the modular curve $X(7)$ .

Theorem(Monodromy of the object of size 24)

Seams fixed by convention (inversion, transvection, rotation, Singer, reversal) are coherent where they meet. The flex-tangent map $\tau$ , a flex going to the other flex on its tangent, has power 4, so the two natural seams between the flexes and their tangents close a cycle with monodromy of order 3. The three role seams into the Heawood matchings have relative powers 1, 2 and 4. Under every seam, $\tau$ is Hall’s multiplier 2 on the labellings and the fourth-power map on $7A$ and $7B$ .

Proposition(Cusps and flexes)

There is a unique isomorphism $\psi\colon X(7)\to X$ intertwining the action of $G$ on the modular curve with Klein’s representation. It maps the cusps onto the flexes and the cusp $\infty$ to $(0:0:1)$ , and it makes the transvection and rotation seams agree. Consequently the tangent to $X(7)$ at the cusp $\infty$ , in its canonical embedding, meets $X(7)$ again at the cusp $2/7$ .

Theorem(The Frobenius at two is coherent)

$\tau$ is the Frobenius at 2 made equivariant. Twisted by $\rho(h)$ , the Frobenius $\zeta\mapsto\zeta^2$ acts on the flexes as $\tau$ , and so does the arithmetic Frobenius on their reductions, the 24 points of the quartic over $\F_8$ . On the coordinatizations of the Fano plane by $\F_8$ , the vertex link of the octonion completion, post-composition with the Frobenius again has power 4, and on both sides the twisting element has power 2.

Example

In Thurston’s congruence link complement it is the cusps with one of the three classes of parallel edges of their cusp torus. Over it the double cover adds the new object of size 48, the nonzero vectors of $\F_7^2$ , with the scalars $\F_7^\times\cong C_6$ as automorphisms. The automorphisms lying over $\tau$ are the scalings by 2, of order 3, and by 5, of order 6 with cube $-I$ : $\tau^3=-I$ is a choice of twisting element, not forced.