LooPindex - 12^3

$12^{3}_{82}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.97614
- on average over minimal pinning sets: 2.4375
- on average over optimal pinning sets: 2.33333

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{2, 4, 7, 8, 9, 11}	6	[2, 2, 2, 2, 3, 3]	2.33
B (optimal)	•	{1, 2, 6, 7, 8, 9}	6	[2, 2, 2, 2, 3, 3]	2.33
C (optimal)	•	{1, 2, 4, 7, 8, 9}	6	[2, 2, 2, 2, 3, 3]	2.33
a (minimal)	•	{2, 3, 5, 6, 7, 8, 9, 11}	8	[2, 2, 2, 2, 3, 3, 4, 4]	2.75

Cardinality	Optimal pinning sets	Minimal suboptimal pinning sets	Nonminimal pinning sets	Averaged mean-degree
6	3	0	0	2.33
7	0	0	16	2.64
8	0	1	35	2.87
9	0	0	42	3.05
10	0	0	26	3.18
11	0	0	8	3.27
12	0	0	1	3.33
Total	3	1	128

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