$12^{1}_{362}$ - Minimal pinning sets
Pinning sets for 12^1_362 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_362 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 12}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,6,7,0],[0,8,9,4],[1,3,9,1],[1,9,6,6],[2,5,5,7],[2,6,8,8],[3,7,7,9],[3,8,5,4]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,9,15,10],[12,19,13,20],[1,17,2,16],[8,15,9,16],[10,3,11,4],[4,11,5,12],[5,18,6,19],[17,6,18,7],[2,7,3,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,4,-16,-5)(20,5,-1,-6)(12,7,-13,-8)(2,9,-3,-10)(18,11,-19,-12)(6,13,-7,-14)(14,19,-15,-20)(3,16,-4,-17)(10,17,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-13,6)(-2,-10,-18,-12,-8)(-3,-17,10)(-4,15,19,11,17)(-5,20,-15)(-6,-14,-20)(-7,12,-19,14)(-9,2)(-11,18)(-16,3,9,1,5)(4,16)(7,13)
Loop annotated with half-edges
12^1_362 annotated with half-edges