$12^{1}_{178}$ - Minimal pinning sets
Pinning sets for 12^1_178 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_178 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 288
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03466
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 6, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 4, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 4, 6, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.4
6
0
0
24
2.69
7
0
0
61
2.9
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,7,7,5],[2,4,8,9],[3,9,9,8],[3,8,4,4],[5,7,6,9],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,19,15,20],[15,12,16,13],[1,16,2,17],[18,5,19,6],[11,4,12,5],[2,9,3,10],[17,7,18,6],[7,10,8,11],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(14,1,-15,-2)(2,13,-3,-14)(18,5,-19,-6)(3,6,-4,-7)(11,8,-12,-9)(16,9,-17,-10)(17,12,-18,-13)(10,15,-11,-16)(4,19,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,-7)(-2,-14)(-4,-20,7)(-5,18,12,8,20)(-6,3,13,-18)(-8,11,15,1)(-9,16,-11)(-10,-16)(-12,17,9)(-13,2,-15,10,-17)(-19,4,6)(5,19)
Loop annotated with half-edges
12^1_178 annotated with half-edges