$12^{2}_{194}$ - Minimal pinning sets
Pinning sets for 12^2_194 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_194 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 9, 10}
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
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,7,8,5],[1,4,2,1],[2,9,9,3],[3,8,4,3],[4,7,9,9],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,12,16,11],[13,10,14,11],[5,19,6,20],[1,18,2,17],[12,17,13,16],[4,9,5,10],[18,6,19,7],[2,7,3,8],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(15,4,-16,-5)(14,5,-1,-6)(2,7,-3,-8)(11,8,-12,-9)(9,18,-10,-19)(19,10,-20,-11)(20,13,-15,-14)(3,16,-4,-17)(12,17,-13,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6)(-2,-8,11,-20,-14,-6)(-3,-17,12,8)(-4,15,13,17)(-5,14,-15)(-7,2)(-9,-19,-11)(-10,19)(-12,-18,9)(-13,20,10,18)(-16,3,7,1,5)(4,16)
Multiloop annotated with half-edges
12^2_194 annotated with half-edges