LooPindex - 12^2

$12^{2}_{261}$ - 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, 3, 4, 5, 6, 7, 10}	7	[2, 2, 2, 2, 2, 3, 3]	2.29
B (optimal)	•	{1, 2, 4, 5, 6, 7, 10}	7	[2, 2, 2, 2, 2, 3, 3]	2.29
C (optimal)	•	{1, 2, 4, 5, 6, 7, 9}	7	[2, 2, 2, 2, 2, 3, 3]	2.29
a (minimal)	•	{1, 3, 4, 5, 6, 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: [[1,1,2,2],[0,3,4,0],[0,5,5,0],[1,6,6,4],[1,3,6,7],[2,7,8,2],[3,9,4,3],[4,9,8,5],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[19,1,20,2],[12,6,13,7],[10,13,11,14],[2,18,3,19],[5,11,6,12],[14,17,15,18],[3,15,4,16],[16,4,17,5]]
Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (10,1,-11,-2)(18,3,-19,-4)(15,4,-16,-5)(13,6,-14,-7)(2,17,-3,-18)(16,19,-17,-20)(8,9,-1,-10)(20,11,-9,-12)(7,12,-8,-13)(5,14,-6,-15)
- 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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-18,-4,15,-6,13,-8,-10)(-3,18)(-5,-15)(-7,-13)(-9,8,12)(-11,20,-17,2)(-12,7,-14,5,-16,-20)(-19,16,4)(1,9,11)(3,17,19)(6,14)