$12^{1}_{224}$ - Minimal pinning sets
Pinning sets for 12^1_224 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_224 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
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: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 5, 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
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,3,3],[0,2,2,6],[0,7,8,8],[1,9,9,7],[1,7,7,3],[4,6,6,5],[4,9,9,4],[5,8,8,5]]
PD code (use to draw this loop with SnapPy): [[17,20,18,1],[5,16,6,17],[6,19,7,20],[18,7,19,8],[1,10,2,11],[13,4,14,5],[15,8,16,9],[9,14,10,15],[2,12,3,11],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(16,5,-17,-6)(13,6,-14,-7)(19,8,-20,-9)(9,20,-10,-1)(1,10,-2,-11)(7,12,-8,-13)(3,14,-4,-15)(4,17,-5,-18)(15,18,-16,-19)
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,-15,-19,-9)(-2,11)(-4,-18,15)(-5,16,18)(-6,13,-8,19,-16)(-7,-13)(-10,1)(-12,7,-14,3)(-17,4,14,6)(-20,9)(2,10,20,8,12)(5,17)
Loop annotated with half-edges
12^1_224 annotated with half-edges