$12^{1}_{175}$ - Minimal pinning sets
Pinning sets for 12^1_175 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_175 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 384
of which optimal: 6
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.1089
on average over minimal pinning sets: 2.66667
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 6, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 4, 5, 7}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 4, 5, 6}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 2, 4, 6, 10}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {1, 2, 3, 4, 6}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 4, 5, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
6
0
0
2.67
6
0
1
34
2.87
7
0
0
86
3.01
8
0
0
115
3.12
9
0
0
90
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
6
1
377
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,7],[0,8,8,9],[0,9,7,5],[1,4,6,6],[1,5,5,2],[2,4,9,8],[3,7,9,3],[3,8,7,4]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,9,7,10],[10,19,11,20],[4,13,5,14],[1,16,2,17],[17,8,18,9],[7,18,8,19],[11,2,12,3],[14,3,15,4],[15,12,16,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(12,1,-13,-2)(18,3,-19,-4)(15,6,-16,-7)(4,7,-5,-8)(13,10,-14,-11)(2,11,-3,-12)(19,14,-20,-15)(5,16,-6,-17)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,18,-9)(-2,-12)(-4,-8,-18)(-5,-17,8)(-6,15,-20,9,17)(-7,4,-19,-15)(-10,13,1)(-11,2,-13)(-14,19,3,11)(-16,5,7)(6,16)(10,20,14)
Loop annotated with half-edges
12^1_175 annotated with half-edges