$12^{1}_{329}$ - Minimal pinning sets
Pinning sets for 12^1_329 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_329 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 304
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03816
on average over minimal pinning sets: 2.47222
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 6, 8, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
0
8
2.58
6
0
2
28
2.78
7
0
0
65
2.93
8
0
0
86
3.06
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
1
2
301
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,7,4],[0,3,8,8],[1,8,6,1],[2,5,9,7],[3,6,9,9],[4,9,5,4],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,13,5,14],[19,12,20,13],[6,12,7,11],[1,11,2,10],[14,3,15,4],[15,18,16,19],[7,16,8,17],[2,9,3,10],[17,8,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(15,6,-16,-7)(20,7,-1,-8)(8,19,-9,-20)(9,4,-10,-5)(17,10,-18,-11)(2,11,-3,-12)(5,14,-6,-15)(13,16,-14,-17)(3,18,-4,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,-19,8)(-2,-12)(-4,9,19)(-5,-15,-7,20,-9)(-6,15)(-8,-20)(-10,17,-14,5)(-11,2,-13,-17)(-16,13,1,7)(-18,3,11)(4,18,10)(6,14,16)
Loop annotated with half-edges
12^1_329 annotated with half-edges