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Each card uses the same representative and vertical direction as all ruling figures in its detail view.

O(4) on P^2

Polygon used for every displayed ruling


SL(2,Z) representative: Conv{(-2,-1), (2,-1), (-2,3)}
vertical direction: (-1,2)

Area 8

Boundary points 12

Interior points 3

Ruling counts by genus
Genus	Total	Saved figures here
3	1	1
2	16	1 symmetry representative
1	104	6 symmetry representatives
0	304	13 symmetry representatives

Ruling polynomial

R_Δ(z) = z⁶ + 16z⁴ + 104z² + 304

Saved ruling figures

Switches are the black points. Each multiplicity below is the size of that figure's symmetry orbit in the fixed degree-four diagram.

Genus 3 ruling 1 total · 1 symmetry representative

O(4) on P^2, genus 3 ruling — Genus 3 · symmetry orbit ×1 By symmetry: 1 ruling

Genus 2 rulings 16 rulings · 1 symmetry representative

O(4) genus-two ruling G2D01 — G2D01 · symmetry orbit ×16D28 · 42 switches · A/B/V 14/0/28

Genus 1 rulings 104 rulings · 6 symmetry representatives

O(4) genus-one ruling G1D01 — G1D01 · symmetry orbit ×16D22 · 34 switches · A/B/V 11/1/22

O(4) genus-one ruling G1D02 — G1D02 · symmetry orbit ×32D24 · 36 switches · A/B/V 12/0/24

O(4) genus-one ruling G1D03 — G1D03 · symmetry orbit ×16D22 · 34 switches · A/B/V 11/1/22

O(4) genus-one ruling G1D04 — G1D04 · symmetry orbit ×16D24 · 36 switches · A/B/V 12/0/24

O(4) genus-one ruling G1D05 — G1D05 · symmetry orbit ×16D24 · 36 switches · A/B/V 12/0/24

O(4) genus-one ruling G1D06 — G1D06 · symmetry orbit ×8D24 · 36 switches · A/B/V 12/0/24

Genus 0 rational rulings 304 rational rulings · 13 symmetry representatives

O(4) rational ruling D01 — D01 · symmetry orbit ×16By symmetry: 16 rulings

O(4) rational ruling D02 — D02 · symmetry orbit ×32By symmetry: 32 rulings

O(4) rational ruling D03 — D03 · symmetry orbit ×32By symmetry: 32 rulings

O(4) rational ruling D04 — D04 · symmetry orbit ×32By symmetry: 32 rulings

O(4) rational ruling D05 — D05 · symmetry orbit ×16By symmetry: 16 rulings

O(4) rational ruling D06 — D06 · symmetry orbit ×32By symmetry: 32 rulings

O(4) rational ruling D07 — D07 · symmetry orbit ×16By symmetry: 16 rulings

O(4) rational ruling D08 — D08 · symmetry orbit ×32By symmetry: 32 rulings

O(4) rational ruling D09 — D09 · symmetry orbit ×32By symmetry: 32 rulings

O(4) rational ruling D10 — D10 · symmetry orbit ×16By symmetry: 16 rulings

O(4) rational ruling D11 — D11 · symmetry orbit ×16By symmetry: 16 rulings

O(4) rational ruling D12 — D12 · symmetry orbit ×16By symmetry: 16 rulings

O(4) annular rational ruling R01 — R01 · annular · symmetry orbit ×16By symmetry: 16 rulings

These figures use direct ruling certificates in the fixed v = (-1,2) chamber. The single genus-two all-disk orbit represents all 16 genus-two rulings, and an exhaustive exact annular search gives zero additional genus-two rulings. The six genus-one all-disk orbits represent all 104 genus-one rulings, again with zero annular rulings. In genus zero, the twelve D-orbits represent 288 all-disk rulings and R01 represents 16 annular rulings.

O(2,4) on P^1 × P^1

Polygon used for every displayed ruling


SL(2,Z) representative: Conv{(0,0), (4,0), (4,2), (0,2)}
source-to-count matrix: [[1,-1], [1,0]]
vertical direction: (1,1)

Area 8

Boundary points 12

Interior points 3

Ruling counts by genus
Genus	Total	Saved figures here
3	1	1
2	16	1 symmetry representative
1	96	7 symmetry representatives
0	256	11 symmetry representatives

Ruling polynomial

R_Δ(z) = z⁶ + 16z⁴ + 96z² + 256

Saved ruling figures

Switches are the black points. Every figure uses the same exact 8-by-4 rectangular Lambda. Each multiplicity is the size of its symmetry orbit in that fixed diagram.

Genus 3 ruling 1 total · 1 symmetry representative

O(2,4) genus-three ruling G3D01 — G3D01 · symmetry orbit ×1D16 · 32 switches · A/B/V 16/0/16

Genus 2 rulings 16 rulings · 1 symmetry representative

O(2,4) genus-two ruling G2D01 — G2D01 · symmetry orbit ×16D14 · 28 switches · A/B/V 14/0/14

Genus 1 rulings 96 rulings · 7 symmetry representatives

O(2,4) genus-one ruling G1D01 — G1D01 · symmetry orbit ×16D12 · 24 switches · A/B/V 12/0/12

O(2,4) genus-one ruling G1D02 — G1D02 · symmetry orbit ×16D12 · 24 switches · A/B/V 12/0/12

O(2,4) genus-one ruling G1D03 — G1D03 · symmetry orbit ×8D12 · 24 switches · A/B/V 12/0/12

O(2,4) genus-one ruling G1D04 — G1D04 · symmetry orbit ×16D12 · 24 switches · A/B/V 12/0/12

O(2,4) genus-one ruling G1D05 — G1D05 · symmetry orbit ×16D10 · 22 switches · A/B/V 10/2/10

O(2,4) genus-one ruling G1D06 — G1D06 · symmetry orbit ×16D12 · 24 switches · A/B/V 12/0/12

O(2,4) genus-one ruling G1D07 — G1D07 · symmetry orbit ×8D12 · 24 switches · A/B/V 12/0/12

Genus 0 rational rulings 256 rational rulings · 11 symmetry representatives

O(2,4) rational ruling D01 — D01 · symmetry orbit ×16D10 · 20 switches · A/B/V 10/0/10

O(2,4) rational ruling D02 — D02 · symmetry orbit ×32D10 · 20 switches · A/B/V 10/0/10

O(2,4) rational ruling D03 — D03 · symmetry orbit ×16D6 · 16 switches · A/B/V 6/4/6

O(2,4) rational ruling D04 — D04 · symmetry orbit ×32D8 · 18 switches · A/B/V 8/2/8

O(2,4) rational ruling D05 — D05 · symmetry orbit ×32D10 · 20 switches · A/B/V 10/0/10

O(2,4) rational ruling D06 — D06 · symmetry orbit ×16D6 · 16 switches · A/B/V 6/4/6

O(2,4) rational ruling D07 — D07 · symmetry orbit ×32D8 · 18 switches · A/B/V 8/2/8

O(2,4) rational ruling D08 — D08 · symmetry orbit ×32D8 · 18 switches · A/B/V 8/2/8

O(2,4) rational ruling D09 — D09 · symmetry orbit ×16D10 · 20 switches · A/B/V 10/0/10

O(2,4) rational ruling D10 — D10 · symmetry orbit ×16D10 · 20 switches · A/B/V 10/0/10

O(2,4) rational ruling D11 — D11 · symmetry orbit ×16D10 · 20 switches · A/B/V 10/0/10

These figures are direct ruling certificates in one fixed, equally spaced 8-by-4 diagram. The complete no-cutoff annular-phase audit gives zero annular rulings in every genus. The displayed symmetry representatives account for all 1, 16, 96, and 256 rulings in genera 3, 2, 1, and 0, respectively.