LooPindex - 12^2

$12^{2}_{112}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.9742
- on average over minimal pinning sets: 2.42262
- on average over optimal pinning sets: 2.33333

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{2, 3, 5, 6, 9, 12}	6	[2, 2, 2, 2, 3, 3]	2.33
a (minimal)	•	{1, 2, 4, 5, 6, 9, 12}	7	[2, 2, 2, 2, 3, 3, 3]	2.43
b (minimal)	•	{2, 3, 5, 6, 8, 9, 11}	7	[2, 2, 2, 2, 3, 3, 3]	2.43
c (minimal)	•	{1, 2, 4, 5, 6, 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	2	6	2.57
8	0	1	23	2.79
9	0	0	34	3.0
10	0	0	24	3.17
11	0	0	8	3.27
12	0	0	1	3.33
Total	1	3	96

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