LooPindex - 12^2

$12^{2}_{2}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.76717
- on average over minimal pinning sets: 2.0
- on average over optimal pinning sets: 2.0

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{1, 2, 3, 4, 5, 6, 9, 11}	8	[2, 2, 2, 2, 2, 2, 2, 2]	2.00

Cardinality	Optimal pinning sets	Nonminimal pinning sets	Averaged mean-degree
8	1	0	2.0
9	0	4	2.44
10	0	6	2.8
11	0	4	3.09
12	0	1	3.33
Total	1	15

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