$12^{1}_{138}$ - Minimal pinning sets
Pinning sets for 12^1_138 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_138 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, 5, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 3, 4, 5, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 3, 4, 5, 7, 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, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,4],[0,5,3,0],[0,2,6,1],[1,7,8,1],[2,9,6,6],[3,5,5,7],[4,6,9,8],[4,7,9,9],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,18,7,17],[4,19,5,20],[1,19,2,18],[7,16,8,17],[10,3,11,4],[2,11,3,12],[12,15,13,16],[8,13,9,14],[14,9,15,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(17,2,-18,-3)(11,6,-12,-7)(19,8,-20,-9)(5,10,-6,-11)(9,12,-10,-13)(4,13,-5,-14)(14,3,-15,-4)(15,18,-16,-19)(1,16,-2,-17)
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,-3,14,-5,-11,-7)(-2,17)(-4,-14)(-6,11)(-8,19,-16,1)(-9,-13,4,-15,-19)(-10,5,13)(-12,9,-20,7)(-18,15,3)(2,16,18)(6,10,12)(8,20)
Loop annotated with half-edges
12^1_138 annotated with half-edges