LooPindex - 12^3

$12^{3}_{66}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.97469
- on average over minimal pinning sets: 2.45833
- on average over optimal pinning sets: 2.33333

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{2, 3, 6, 7, 8, 9}	6	[2, 2, 2, 2, 3, 3]	2.33
a (minimal)	•	{1, 2, 4, 5, 6, 7, 8, 9}	8	[2, 2, 2, 2, 3, 3, 3, 3]	2.50
b (minimal)	•	{1, 4, 5, 6, 7, 8, 9, 11}	8	[2, 2, 2, 2, 3, 3, 3, 3]	2.50
c (minimal)	•	{1, 3, 4, 6, 7, 8, 9, 11}	8	[2, 2, 2, 2, 3, 3, 3, 3]	2.50

Cardinality	Optimal pinning sets	Minimal suboptimal pinning sets	Nonminimal pinning sets	Averaged mean-degree
6	1	0	0	2.33
7	0	0	6	2.62
8	0	3	15	2.78
9	0	0	28	2.97
10	0	0	22	3.15
11	0	0	8	3.27
12	0	0	1	3.33
Total	1	3	80

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