$12^{1}_{361}$ - Minimal pinning sets
Pinning sets for 12^1_361 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_361 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, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 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, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,7],[0,7,7,4],[1,3,8,5],[1,4,8,6],[2,5,9,2],[2,9,3,3],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,4,11,3],[1,15,2,14],[18,13,19,14],[7,12,8,13],[4,12,5,11],[2,15,3,16],[6,17,7,18],[5,17,6,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,8,-2,-9)(17,6,-18,-7)(9,2,-10,-3)(5,10,-6,-11)(16,11,-17,-12)(12,19,-13,-20)(20,13,-1,-14)(14,3,-15,-4)(4,15,-5,-16)(7,18,-8,-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,-9,-3,14)(-2,9)(-4,-16,-12,-20,-14)(-5,-11,16)(-6,17,11)(-7,-19,12,-17)(-8,1,13,19)(-10,5,15,3)(-13,20)(-15,4)(-18,7)(2,8,18,6,10)
Loop annotated with half-edges
12^1_361 annotated with half-edges