$12^{1}_{255}$ - Minimal pinning sets
Pinning sets for 12^1_255 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_255 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, 2, 4, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 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, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,6],[0,7,7,4],[0,3,7,5],[1,4,8,1],[2,8,9,2],[3,9,4,3],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[13,6,14,7],[14,19,15,20],[8,3,9,4],[1,4,2,5],[5,12,6,13],[18,15,19,16],[2,9,3,10],[11,16,12,17],[17,10,18,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(18,1,-19,-2)(10,3,-11,-4)(19,6,-20,-7)(14,7,-15,-8)(2,9,-3,-10)(15,12,-16,-13)(8,13,-9,-14)(11,16,-12,-17)(4,17,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-5)(-2,-10,-4,-18)(-3,10)(-6,19,1)(-7,14,-9,2,-19)(-8,-14)(-11,-17,4)(-12,15,7,-20,5,17)(-13,8,-15)(-16,11,3,9,13)(6,20)(12,16)
Loop annotated with half-edges
12^1_255 annotated with half-edges