$12^{2}_{96}$ - Minimal pinning sets
Pinning sets for 12^2_96 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_96 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 384
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: 3.03466
on average over minimal pinning sets: 2.25
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 5, 9, 12}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {2, 5, 8, 9}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
15
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
0
382
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,4],[0,5,6,1],[2,7,5,2],[3,4,8,6],[3,5,9,7],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[10,19,11,20],[1,6,2,7],[18,11,19,12],[5,17,6,18],[2,17,3,16],[12,16,13,15],[4,14,5,15],[3,14,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(14,5,-15,-6)(11,16,-12,-17)(18,1,-19,-2)(2,19,-3,-20)(20,17,-9,-18)(9,8,-10,-1)(3,10,-4,-11)(15,12,-16,-13)(6,13,-7,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-9)(-2,-20,-18)(-3,-11,-17,20)(-4,7,13,-16,11)(-5,14,-7)(-6,-14)(-8,9,17,-12,15,5)(-10,3,19,1)(-13,6,-15)(-19,2)(4,10,8)(12,16)
Multiloop annotated with half-edges
12^2_96 annotated with half-edges