$12^{1}_{66}$ - Minimal pinning sets
Pinning sets for 12^1_66 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_66 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
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)
• {1, 2, 3, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,2],[0,1,5,0],[0,6,7,4],[1,3,5,1],[2,4,8,6],[3,5,9,7],[3,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,9,13,10],[10,19,11,20],[1,15,2,14],[8,13,9,14],[18,7,19,8],[15,7,16,6],[2,6,3,5],[17,4,18,5],[16,4,17,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,14,-2,-15)(15,2,-16,-3)(3,20,-4,-1)(4,13,-5,-14)(16,5,-17,-6)(10,7,-11,-8)(18,9,-19,-10)(6,11,-7,-12)(12,19,-13,-20)(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,-15,-3)(-2,15)(-4,-14,1)(-5,16,2,14)(-6,-12,-20,3,-16)(-7,10,-19,12)(-8,-18,-10)(-9,18)(-11,6,-17,8)(-13,4,20)(5,13,19,9,17)(7,11)
Loop annotated with half-edges
12^1_66 annotated with half-edges