$12^{1}_{65}$ - Minimal pinning sets
Pinning sets for 12^1_65 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_65 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 320
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10067
on average over minimal pinning sets: 2.63333
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 7, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {2, 3, 6, 7, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 4, 6, 7, 9}
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
3
0
0
2.6
6
0
3
19
2.8
7
0
0
64
2.97
8
0
0
97
3.1
9
0
0
83
3.2
10
0
0
40
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
3
3
314
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6]
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],[3,4,8,6],[3,5,8,9],[4,9,9,8],[5,7,9,6],[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],[2,15,3,16],[17,4,18,5],[6,3,7,4],[16,6,17,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(15,2,-16,-3)(3,14,-4,-15)(12,5,-13,-6)(19,6,-20,-7)(10,7,-11,-8)(18,9,-19,-10)(20,13,-1,-14)(11,16,-12,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,14)(-2,15,-4)(-3,-15)(-5,12,16,2)(-6,19,9,17,-12)(-7,10,-19)(-8,-18,-10)(-9,18)(-11,-17,8)(-13,20,6)(-14,3,-16,11,7,-20)(1,13,5)
Loop annotated with half-edges
12^1_65 annotated with half-edges