LooPindex - 11^2

$11^{2}_{30}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.80286
- 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, 4, 5, 6, 7, 8}	7	[2, 2, 2, 2, 2, 2, 3]	2.14
a (minimal)	•	{1, 3, 4, 5, 6, 7, 8, 10}	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	4	2.48
9	0	0	8	2.83
10	0	0	5	3.12
11	0	0	1	3.27
Total	1	1	18

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