$12^{1}_{177}$ - Minimal pinning sets
Pinning sets for 12^1_177 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_177 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 280
of which optimal: 3
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03877
on average over minimal pinning sets: 2.50667
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, 2, 3, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 6, 7, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {1, 2, 4, 5, 7, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
2
19
2.68
7
0
0
58
2.89
8
0
0
84
3.04
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
3
2
275
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,7,5],[2,4,8,8],[3,9,9,7],[4,6,9,8],[5,7,9,5],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,19,9,20],[9,6,10,7],[1,10,2,11],[11,18,12,19],[12,5,13,6],[2,16,3,15],[17,14,18,15],[4,13,5,14],[16,4,17,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,14,-2,-15)(10,3,-11,-4)(7,4,-8,-5)(18,5,-19,-6)(19,8,-20,-9)(2,11,-3,-12)(15,12,-16,-13)(13,20,-14,-1)(9,16,-10,-17)(6,17,-7,-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,-15,-13)(-2,-12,15)(-3,10,16,12)(-4,7,17,-10)(-5,18,-7)(-6,-18)(-8,19,5)(-9,-17,6,-19)(-11,2,14,20,8,4)(-14,1)(-16,9,-20,13)(3,11)
Loop annotated with half-edges
12^1_177 annotated with half-edges