$12^{2}_{215}$ - Minimal pinning sets
Pinning sets for 12^2_215 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_215 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
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, 3, 7, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,8,8,9],[2,9,9,6],[3,5,9,7],[3,6,8,8],[4,7,7,4],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[12,19,13,20],[1,13,2,14],[16,8,17,9],[4,18,5,19],[2,5,3,6],[14,6,15,7],[7,15,8,16],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(9,2,-10,-3)(18,3,-19,-4)(20,5,-11,-6)(15,6,-16,-7)(4,19,-5,-20)(1,12,-2,-13)(16,13,-17,-14)(7,14,-8,-15)(8,17,-9,-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,-13,16,6,-11)(-2,9,17,13)(-3,18,-9)(-4,-20,-6,15,-8,-18)(-5,20)(-7,-15)(-10,11,5,19,3)(-12,1)(-14,7,-16)(-17,8,14)(-19,4)(2,12,10)
Multiloop annotated with half-edges
12^2_215 annotated with half-edges