$12^{1}_{63}$ - Minimal pinning sets
Pinning sets for 12^1_63 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_63 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 240
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03455
on average over minimal pinning sets: 2.45
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, 3, 4, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 5, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2.4
6
0
2
13
2.66
7
0
0
45
2.86
8
0
0
71
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
236
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,4,0],[0,5,6,1],[1,7,5,2],[3,4,8,8],[3,8,9,7],[4,6,9,9],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,9,13,10],[10,19,11,20],[1,8,2,9],[13,18,14,19],[14,7,15,8],[2,6,3,5],[17,4,18,5],[6,15,7,16],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(1,18,-2,-19)(10,3,-11,-4)(17,4,-18,-5)(8,5,-9,-6)(16,7,-17,-8)(2,11,-3,-12)(19,12,-20,-13)(9,14,-10,-15)(6,15,-7,-16)
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,-19,-13)(-2,-12,19)(-3,10,14,20,12)(-4,17,7,15,-10)(-5,8,-17)(-6,-16,-8)(-7,16)(-9,-15,6)(-11,2,18,4)(-14,9,5,-18,1)(-20,13)(3,11)
Loop annotated with half-edges
12^1_63 annotated with half-edges