$12^{2}_{67}$ - Minimal pinning sets
Pinning sets for 12^2_67 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_67 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
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
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,3],[0,2,8,8],[0,6,5,5],[1,4,4,1],[1,4,9,2],[2,9,9,8],[3,7,9,3],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,3,10,4],[13,7,14,8],[14,19,15,20],[1,12,2,11],[2,10,3,11],[4,12,5,13],[6,16,7,17],[18,15,19,16],[5,18,6,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-9,-2)(12,5,-13,-6)(8,9,-1,-10)(10,7,-11,-8)(2,11,-3,-12)(17,14,-18,-15)(4,15,-5,-16)(16,3,-17,-4)(13,18,-14,-19)(6,19,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,-7,10)(-2,-12,-6,-20)(-3,16,-5,12)(-4,-16)(-8,-10)(-9,8,-11,2)(-13,-19,6)(-14,17,3,11,7,19)(-15,4,-17)(-18,13,5,15)(1,9)(14,18)
Multiloop annotated with half-edges
12^2_67 annotated with half-edges