$12^{2}_{262}$ - Minimal pinning sets
Pinning sets for 12^2_262 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_262 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 112
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 6, 7, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 6, 7, 11, 12}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,6],[0,6,6,0],[1,7,5,1],[2,4,7,8],[2,8,3,3],[4,9,9,5],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,16,10,17],[17,7,18,8],[19,1,20,2],[15,10,16,11],[6,14,7,15],[18,3,19,2],[11,5,12,6],[13,3,14,4],[4,12,5,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(15,2,-16,-3)(17,4,-18,-5)(5,16,-6,-17)(11,6,-12,-7)(7,10,-8,-11)(1,12,-2,-13)(19,14,-20,-15)(3,18,-4,-19)(13,20,-14,-9)
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,-9)(-2,15,-20,13)(-3,-19,-15)(-4,17,-6,11,-8,9,-14,19)(-5,-17)(-7,-11)(-10,7,-12,1)(-16,5,-18,3)(2,12,6,16)(4,18)(8,10)(14,20)
Multiloop annotated with half-edges
12^2_262 annotated with half-edges