$12^{1}_{308}$ - Minimal pinning sets
Pinning sets for 12^1_308 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_308 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 48
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: 2.86152
on average over minimal pinning sets: 2.14286
on average over optimal pinning sets: 2.14286
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 8, 9, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
B (optimal)
• {1, 2, 4, 6, 8, 9, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
2
0
0
2.14
8
0
0
9
2.53
9
0
0
16
2.82
10
0
0
14
3.04
11
0
0
6
3.21
12
0
0
1
3.33
Total
2
0
46
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,6,0],[1,7,2,1],[2,8,8,9],[3,9,9,3],[4,9,8,8],[5,7,7,5],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,17,9,18],[19,16,20,17],[10,2,11,1],[18,7,19,8],[15,4,16,5],[2,12,3,11],[13,6,14,7],[5,14,6,15],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(10,3,-11,-4)(15,4,-16,-5)(8,19,-9,-20)(20,9,-1,-10)(2,11,-3,-12)(16,13,-17,-14)(5,14,-6,-15)(6,17,-7,-18)(18,7,-19,-8)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,10)(-2,-12)(-4,15,-6,-18,-8,-20,-10)(-5,-15)(-7,18)(-9,20)(-11,2,-13,16,4)(-14,5,-16)(-17,6,14)(-19,8)(1,9,19,7,17,13)(3,11)
Loop annotated with half-edges
12^1_308 annotated with half-edges