$12^{1}_{214}$ - Minimal pinning sets
Pinning sets for 12^1_214 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_214 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97193
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 3, 4, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.33
7
0
0
20
2.66
8
0
0
41
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,7,8,8],[1,8,7,1],[1,9,6,2],[2,5,9,9],[3,9,4,8],[3,7,4,3],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,11,5,12],[19,6,20,7],[1,14,2,15],[12,3,13,4],[10,7,11,8],[18,9,19,10],[13,16,14,17],[2,16,3,15],[8,17,9,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(6,3,-7,-4)(14,5,-15,-6)(20,7,-1,-8)(17,10,-18,-11)(11,18,-12,-19)(9,12,-10,-13)(4,13,-5,-14)(1,16,-2,-17)(8,19,-9,-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,-17,-11,-19,8)(-2,15,5,13,-10,17)(-3,6,-15)(-4,-14,-6)(-5,14)(-7,20,-9,-13,4)(-8,-20)(-12,9,19)(-16,1,7,3)(-18,11)(2,16)(10,12,18)
Loop annotated with half-edges
12^1_214 annotated with half-edges