$12^{1}_{112}$ - Minimal pinning sets
Pinning sets for 12^1_112 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_112 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 480
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0461
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 5, 11}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {1, 3, 7, 11}
4
[2, 2, 2, 4]
2.50
D (optimal)
• {1, 3, 9, 11}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
4
0
0
2.38
5
0
0
26
2.68
6
0
0
74
2.87
7
0
0
121
3.01
8
0
0
125
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
476
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,7,7,8],[0,5,1,1],[1,4,8,2],[2,8,9,9],[2,9,3,3],[3,9,6,5],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[9,2,10,3],[19,6,20,7],[4,16,5,15],[1,8,2,9],[10,8,11,7],[18,13,19,14],[5,16,6,17],[14,11,15,12],[12,17,13,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(17,4,-18,-5)(5,8,-6,-9)(19,6,-20,-7)(12,9,-13,-10)(16,11,-17,-12)(13,2,-14,-3)(3,14,-4,-15)(10,15,-11,-16)(1,18,-2,-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,-19,-7)(-2,13,9,-6,19)(-3,-15,10,-13)(-4,17,11,15)(-5,-9,12,-17)(-8,5,-18,1)(-10,-16,-12)(-11,16)(-14,3)(-20,7)(2,18,4,14)(6,8,20)
Loop annotated with half-edges
12^1_112 annotated with half-edges