$12^{1}_{84}$ - Minimal pinning sets
Pinning sets for 12^1_84 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_84 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, 3, 5, 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, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,4,5,6],[0,6,7,1],[1,5,5,2],[2,4,4,8],[2,9,7,3],[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],[12,19,13,20],[10,7,11,8],[1,18,2,19],[13,9,14,8],[14,9,15,10],[17,6,18,7],[2,6,3,5],[15,5,16,4],[16,3,17,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(2,13,-3,-14)(3,20,-4,-1)(15,4,-16,-5)(8,5,-9,-6)(6,11,-7,-12)(12,7,-13,-8)(18,9,-19,-10)(19,16,-20,-17)(10,17,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3)(-2,-14)(-4,15,1)(-5,8,-13,2,-15)(-6,-12,-8)(-7,12)(-9,18,-11,6)(-10,-18)(-16,19,9,5)(-17,10,-19)(-20,3,13,7,11,17)(4,20,16)
Loop annotated with half-edges
12^1_84 annotated with half-edges