$12^{1}_{219}$ - Minimal pinning sets
Pinning sets for 12^1_219 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_219 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, 3, 4, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 5, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 4, 5}
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, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,8,8,4],[0,3,9,6],[1,6,6,1],[2,5,5,4],[2,9,9,8],[3,7,9,3],[4,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,6,17,5],[19,4,20,5],[14,9,15,10],[1,9,2,8],[6,18,7,17],[7,18,8,19],[12,3,13,4],[10,13,11,14],[2,11,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(17,2,-18,-3)(3,14,-4,-15)(6,19,-7,-20)(12,7,-13,-8)(20,9,-1,-10)(8,11,-9,-12)(18,13,-19,-14)(15,4,-16,-5)(5,16,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,17,-6,-20,-10)(-3,-15,-5,-17)(-4,15)(-7,12,-9,20)(-8,-12)(-11,8,-13,18,2)(-14,3,-18)(-16,5)(-19,6,16,4,14)(1,9,11)(7,19,13)
Loop annotated with half-edges
12^1_219 annotated with half-edges