$12^{1}_{276}$ - Minimal pinning sets
Pinning sets for 12^1_276 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_276 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 4, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,6],[1,7,7,8],[2,8,8,6],[3,5,9,3],[4,9,9,4],[4,9,5,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[11,8,12,9],[1,12,2,13],[5,18,6,19],[7,14,8,15],[2,14,3,13],[17,4,18,5],[6,16,7,15],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,3,-7,-4)(15,4,-16,-5)(16,7,-17,-8)(8,17,-9,-18)(2,9,-3,-10)(10,1,-11,-2)(18,11,-19,-12)(20,13,-1,-14)(5,14,-6,-15)(12,19,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,6,14)(-2,-10)(-4,15,-6)(-5,-15)(-7,16,4)(-8,-18,-12,-20,-14,5,-16)(-9,2,-11,18)(-13,20)(-17,8)(-19,12)(1,13,19,11)(3,9,17,7)
Loop annotated with half-edges
12^1_276 annotated with half-edges