$12^{1}_{353}$ - Minimal pinning sets
Pinning sets for 12^1_353 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_353 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 128
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98094
on average over minimal pinning sets: 2.39881
on average over optimal pinning sets: 2.38889
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
C (optimal)
• {1, 2, 3, 4, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 4, 5, 8, 9, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.39
7
0
1
15
2.67
8
0
0
35
2.88
9
0
0
40
3.05
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
1
124
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,7],[0,8,8,1],[1,6,6,2],[3,5,5,7],[3,6,9,9],[4,9,9,4],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[14,7,15,8],[19,6,20,7],[16,6,17,5],[1,9,2,8],[18,13,19,14],[17,13,18,12],[4,11,5,12],[9,3,10,2],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(8,1,-9,-2)(3,14,-4,-15)(17,4,-18,-5)(12,5,-13,-6)(6,19,-7,-20)(2,9,-3,-10)(10,15,-11,-16)(16,11,-17,-12)(13,18,-14,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-10,-16,-12,-6,-20,-8)(-3,-15,10)(-4,17,11,15)(-5,12,-17)(-7,20)(-9,2)(-11,16)(-13,-19,6)(-14,3,9,1,7,19)(-18,13,5)(4,14,18)
Loop annotated with half-edges
12^1_353 annotated with half-edges