$12^{1}_{91}$ - Minimal pinning sets
Pinning sets for 12^1_91 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_91 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,6,7,7],[1,7,7,8],[2,8,9,9],[2,9,8,3],[3,4,4,3],[4,6,9,5],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[19,10,20,11],[12,10,13,9],[1,16,2,17],[3,18,4,19],[13,7,14,6],[15,8,16,9],[2,18,3,17],[4,8,5,7],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,3,-13,-4)(4,11,-5,-12)(5,2,-6,-3)(13,6,-14,-7)(7,10,-8,-11)(17,8,-18,-9)(19,14,-20,-15)(15,20,-16,-1)(1,16,-2,-17)(9,18,-10,-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,-17,-9,-19,-15)(-2,5,11,-8,17)(-3,12,-5)(-4,-12)(-6,13,3)(-7,-11,4,-13)(-10,7,-14,19)(-16,1)(-18,9)(-20,15)(2,16,20,14,6)(8,10,18)
Loop annotated with half-edges
12^1_91 annotated with half-edges