LooPindex - 12^1

$12^{1}_{152}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.8647
- on average over minimal pinning sets: 2.19643
- on average over optimal pinning sets: 2.14286

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{1, 2, 3, 4, 5, 8, 11}	7	[2, 2, 2, 2, 2, 2, 3]	2.14
a (minimal)	•	{1, 2, 3, 4, 5, 6, 7, 11}	8	[2, 2, 2, 2, 2, 2, 3, 3]	2.25

Cardinality	Optimal pinning sets	Minimal suboptimal pinning sets	Nonminimal pinning sets	Averaged mean-degree
7	1	0	0	2.14
8	0	1	5	2.46
9	0	0	13	2.76
10	0	0	13	3.02
11	0	0	6	3.21
12	0	0	1	3.33
Total	1	1	38

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