$12^{2}_{165}$ - Minimal pinning sets
Pinning sets for 12^2_165 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_165 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 2, 2]
2.00
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
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,5],[0,6,7,0],[1,7,5,1],[2,4,8,2],[3,9,9,7],[3,6,8,4],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[16,7,1,8],[8,17,9,20],[15,19,16,20],[6,1,7,2],[17,10,18,9],[18,14,19,15],[2,12,3,11],[5,10,6,11],[13,4,14,5],[12,4,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,10,-2,-11)(11,2,-12,-3)(14,5,-15,-6)(7,18,-8,-19)(3,8,-4,-9)(9,16,-10,-1)(19,12,-20,-13)(4,15,-5,-16)(13,20,-14,-17)(17,6,-18,-7)
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,-11,-3,-9)(-2,11)(-4,-16,9)(-5,14,20,12,2,10,16)(-6,17,-14)(-7,-19,-13,-17)(-8,3,-12,19)(-10,1)(-15,4,8,18,6)(-18,7)(-20,13)(5,15)
Multiloop annotated with half-edges
12^2_165 annotated with half-edges