$12^{2}_{191}$ - Minimal pinning sets
Pinning sets for 12^2_191 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_191 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 376
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04011
on average over minimal pinning sets: 2.49
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 7, 12}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 4, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 3, 5]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
4
8
2.57
6
0
0
46
2.8
7
0
0
90
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
4
371
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,7,8],[2,8,9,9],[3,9,7,7],[4,6,6,8],[4,7,9,5],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,5,8,6],[8,19,9,20],[1,9,2,10],[10,4,11,5],[18,13,19,14],[2,17,3,16],[3,15,4,16],[11,15,12,14],[12,17,13,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(12,3,-13,-4)(5,14,-6,-15)(9,18,-10,-19)(19,16,-20,-17)(1,20,-2,-7)(7,6,-8,-1)(17,8,-18,-9)(13,10,-14,-11)(4,11,-5,-12)
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,-7)(-2,15,-6,7)(-3,12,-5,-15)(-4,-12)(-8,17,-20,1)(-9,-19,-17)(-10,13,3,-16,19)(-11,4,-13)(-14,5,11)(-18,9)(2,20,16)(6,14,10,18,8)
Multiloop annotated with half-edges
12^2_191 annotated with half-edges