$12^{1}_{27}$ - Minimal pinning sets
Pinning sets for 12^1_27 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_27 Pinning data
Pinning number of this loop: 3
Total number of pinning sets: 512
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03436
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 5, 11}
3
[2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
3
1
0
0
2.0
4
0
0
9
2.44
5
0
0
36
2.71
6
0
0
84
2.89
7
0
0
126
3.02
8
0
0
126
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
0
511
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: [[1,2,3,4],[0,4,4,5],[0,6,7,7],[0,8,9,5],[0,5,1,1],[1,4,3,6],[2,5,9,7],[2,6,8,2],[3,7,9,9],[3,8,8,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,9,3,10],[19,14,20,15],[4,7,5,8],[1,11,2,10],[11,8,12,9],[15,12,16,13],[13,18,14,19],[6,17,7,18],[5,17,6,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (2,5,-3,-6)(9,6,-10,-7)(20,7,-1,-8)(8,19,-9,-20)(13,10,-14,-11)(17,12,-18,-13)(14,3,-15,-4)(4,15,-5,-16)(11,16,-12,-17)(1,18,-2,-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,-19,8)(-2,-6,9,19)(-3,14,10,6)(-4,-16,11,-14)(-5,2,18,12,16)(-7,20,-9)(-8,-20)(-10,13,-18,1,7)(-11,-17,-13)(-12,17)(-15,4)(3,5,15)
Loop annotated with half-edges
12^1_27 annotated with half-edges