$12^{2}_{101}$ - Minimal pinning sets
Pinning sets for 12^2_101 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_101 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 112
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: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 4, 6, 9, 12}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,7,7],[2,8,8,6],[2,5,9,9],[4,9,8,4],[5,7,9,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[2,9,3,10],[15,4,16,5],[1,11,2,10],[11,8,12,9],[5,17,6,20],[14,19,15,20],[7,12,8,13],[17,7,18,6],[18,13,19,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(4,17,-5,-18)(9,6,-10,-7)(16,7,-1,-8)(8,15,-9,-16)(5,10,-6,-11)(18,11,-19,-12)(1,14,-2,-15)(12,19,-13,-20)(20,3,-17,-4)
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,-15,8)(-2,13,19,11,-6,9,15)(-3,20,-13)(-4,-18,-12,-20)(-5,-11,18)(-7,16,-9)(-8,-16)(-10,5,17,3,-14,1,7)(-17,4)(-19,12)(2,14)(6,10)
Multiloop annotated with half-edges
12^2_101 annotated with half-edges