$12^{1}_{104}$ - Minimal pinning sets
Pinning sets for 12^1_104 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_104 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 4, 6, 9, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 4, 8, 9, 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
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,7],[0,5,1,1],[1,4,8,2],[2,8,8,9],[3,9,9,3],[5,9,6,6],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[15,2,16,3],[19,12,20,13],[4,12,5,11],[1,14,2,15],[16,14,17,13],[18,7,19,8],[5,10,6,11],[17,9,18,8],[9,6,10,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(12,1,-13,-2)(2,11,-3,-12)(19,4,-20,-5)(5,18,-6,-19)(13,6,-14,-7)(7,10,-8,-11)(15,8,-16,-9)(17,14,-18,-15)(9,16,-10,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3)(-2,-12)(-4,19,-6,13,1)(-5,-19)(-7,-11,2,-13)(-8,15,-18,5,-20,3,11)(-9,-17,-15)(-10,7,-14,17)(-16,9)(4,20)(6,18,14)(8,10,16)
Loop annotated with half-edges
12^1_104 annotated with half-edges