LooPindex - 12^3

$12^{3}_{64}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.82247
- on average over minimal pinning sets: 2.16667
- on average over optimal pinning sets: 2.16667

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{1, 2, 3, 4, 5, 6, 7, 11}	8	[2, 2, 2, 2, 2, 2, 2, 3]	2.12
B (optimal)	•	{1, 2, 3, 4, 5, 7, 10, 11}	8	[2, 2, 2, 2, 2, 2, 2, 4]	2.25
C (optimal)	•	{1, 2, 3, 4, 5, 7, 9, 11}	8	[2, 2, 2, 2, 2, 2, 2, 3]	2.12

Cardinality	Optimal pinning sets	Nonminimal pinning sets	Averaged mean-degree
8	3	0	2.17
9	0	9	2.64
10	0	10	2.96
11	0	5	3.16
12	0	1	3.33
Total	3	25

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