$12^{2}_{111}$ - Minimal pinning sets
Pinning sets for 12^2_111 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_111 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
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: 2.99652
on average over minimal pinning sets: 2.4419
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 6, 7, 12}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {2, 3, 6, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 5, 6, 7}
6
[2, 2, 2, 2, 3, 3]
2.33
c (minimal)
• {1, 2, 3, 6, 7, 8, 11}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
d (minimal)
• {2, 3, 4, 5, 6, 7, 9}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
2
7
2.59
7
0
2
31
2.77
8
0
0
58
2.96
9
0
0
54
3.1
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
4
187
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,6,0],[0,6,7,1],[1,7,8,5],[1,4,9,9],[2,7,3,2],[3,6,8,4],[4,7,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,17,12,16],[19,9,20,10],[1,18,2,17],[12,7,13,6],[15,5,16,6],[8,18,9,19],[2,8,3,7],[13,3,14,4],[4,14,5,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(13,2,-14,-3)(3,20,-4,-11)(4,9,-5,-10)(17,6,-18,-7)(1,12,-2,-13)(14,19,-15,-20)(15,8,-16,-9)(5,16,-6,-17)(7,18,-8,-19)
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,-13,-3,-11)(-2,13)(-4,-10,11)(-5,-17,-7,-19,14,2,12,10)(-6,17)(-8,15,19)(-9,4,20,-15)(-12,1)(-14,-20,3)(-16,5,9)(-18,7)(6,16,8,18)
Multiloop annotated with half-edges
12^2_111 annotated with half-edges