LooPindex - 12^2

$12^{2}_{250}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.92126
- on average over minimal pinning sets: 2.33929
- on average over optimal pinning sets: 2.28571

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{1, 2, 3, 5, 6, 7, 10}	7	[2, 2, 2, 2, 2, 3, 3]	2.29
B (optimal)	•	{1, 2, 3, 4, 5, 7, 10}	7	[2, 2, 2, 2, 2, 3, 3]	2.29
C (optimal)	•	{1, 2, 3, 5, 6, 7, 12}	7	[2, 2, 2, 2, 2, 3, 3]	2.29
a (minimal)	•	{1, 2, 3, 4, 5, 7, 9, 12}	8	[2, 2, 2, 2, 2, 3, 3, 4]	2.50

Cardinality	Optimal pinning sets	Minimal suboptimal pinning sets	Nonminimal pinning sets	Averaged mean-degree
7	3	0	0	2.29
8	0	1	13	2.62
9	0	0	24	2.91
10	0	0	19	3.12
11	0	0	7	3.25
12	0	0	1	3.33
Total	3	1	64

Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,6,0],[1,7,2,1],[2,7,8,8],[3,9,9,3],[4,9,8,5],[5,7,9,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,8,12,7],[9,6,10,7],[19,1,20,2],[8,13,9,12],[5,15,6,16],[2,18,3,19],[13,17,14,16],[14,4,15,5],[17,3,18,4]]
Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (20,1,-11,-2)(17,2,-18,-3)(7,4,-8,-5)(5,14,-6,-15)(12,9,-13,-10)(3,18,-4,-19)(19,16,-20,-17)(10,11,-1,-12)(8,13,-9,-14)(15,6,-16,-7)
- Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,16,6,14,-9,12)(-2,17,-20)(-3,-19,-17)(-4,7,-16,19)(-5,-15,-7)(-6,15)(-8,-14,5)(-10,-12)(-11,10,-13,8,4,18,2)(-18,3)(1,11)(9,13)