$12^{2}_{198}$ - Minimal pinning sets
Pinning sets for 12^2_198 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_198 Pinning data
Pinning number of this multiloop: 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: 2.97944
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 5, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 4, 5, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 3, 4, 5, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
D (optimal)
• {1, 2, 3, 4, 5, 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
2.38
7
0
0
20
2.68
8
0
0
41
2.9
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 multiloop 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: [[1,1,2,3],[0,4,4,0],[0,5,6,3],[0,2,7,8],[1,5,5,1],[2,4,4,8],[2,9,9,7],[3,6,9,8],[3,7,9,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[14,19,15,20],[1,15,2,16],[8,12,9,13],[13,7,14,8],[4,18,5,19],[2,5,3,6],[16,6,17,7],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-11,-2)(10,3,-1,-4)(19,4,-20,-5)(16,5,-17,-6)(14,7,-15,-8)(9,18,-10,-19)(2,11,-3,-12)(17,12,-18,-13)(8,13,-9,-14)(6,15,-7,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,4)(-2,-12,17,5,-20)(-3,10,18,12)(-4,19,-10)(-5,16,-7,14,-9,-19)(-6,-16)(-8,-14)(-11,2)(-13,8,-15,6,-17)(-18,9,13)(1,3,11)(7,15)
Multiloop annotated with half-edges
12^2_198 annotated with half-edges