$12^{1}_{206}$ - Minimal pinning sets
Pinning sets for 12^1_206 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_206 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 288
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: 3.03466
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 3, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.4
6
0
0
24
2.69
7
0
0
61
2.9
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,7,7,8],[2,8,9,9],[3,9,8,7],[3,6,4,4],[4,6,9,5],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,19,15,20],[15,12,16,13],[1,16,2,17],[18,7,19,8],[4,11,5,12],[2,10,3,9],[17,9,18,8],[3,6,4,7],[10,5,11,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(1,4,-2,-5)(16,5,-17,-6)(6,15,-7,-16)(7,20,-8,-1)(2,9,-3,-10)(13,10,-14,-11)(18,11,-19,-12)(19,14,-20,-15)(12,17,-13,-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,-5,16,-7)(-2,-10,13,17,5)(-3,8,20,14,10)(-4,1,-8)(-6,-16)(-9,2,4)(-11,18,-13)(-12,-18)(-14,19,11)(-15,6,-17,12,-19)(-20,7,15)(3,9)
Loop annotated with half-edges
12^1_206 annotated with half-edges