LooPindex - 12^2

$12^{2}_{113}$ - 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)	•	{1, 3, 4, 6, 9, 10}	6	[2, 2, 2, 2, 3, 3]	2.33
a (minimal)	•	{1, 3, 4, 6, 8, 9, 11}	7	[2, 2, 2, 2, 3, 3, 3]	2.43
b (minimal)	•	{1, 2, 3, 5, 6, 9, 10}	7	[2, 2, 2, 2, 3, 3, 3]	2.43
c (minimal)	•	{1, 2, 3, 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],[1,6,7,1],[2,8,8,3],[4,8,9,7],[4,6,9,9],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[19,11,20,12],[1,11,2,10],[14,8,15,9],[18,2,19,3],[4,7,5,8],[15,5,16,6],[3,17,4,18],[6,16,7,17]]
Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (13,12,-14,-1)(18,3,-19,-4)(2,5,-3,-6)(17,6,-18,-7)(7,16,-8,-17)(8,11,-9,-12)(14,9,-15,-10)(4,19,-5,-20)(1,20,-2,-13)(10,15,-11,-16)
- 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)(-2,-6,17,-8,-12,13)(-3,18,6)(-4,-20,1,-14,-10,-16,7,-18)(-5,2,20)(-7,-17)(-9,14,12)(-11,8,16)(-15,10)(-19,4)(3,5,19)(9,11,15)