$12^{1}_{82}$ - Minimal pinning sets
Pinning sets for 12^1_82 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_82 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 176
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: 2.97631
on average over minimal pinning sets: 2.34444
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 7, 8}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 5, 7, 9}
6
[2, 2, 2, 2, 3, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,7],[2,8,6,3],[3,5,8,4],[4,9,9,4],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[19,10,20,11],[12,10,13,9],[1,18,2,19],[13,7,14,6],[17,8,18,9],[2,8,3,7],[14,5,15,6],[16,3,17,4],[4,15,5,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(12,3,-13,-4)(4,11,-5,-12)(5,20,-6,-1)(15,6,-16,-7)(7,10,-8,-11)(17,8,-18,-9)(2,13,-3,-14)(19,16,-20,-17)(9,18,-10,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,12,-5)(-2,-14)(-4,-12)(-6,15,1)(-7,-11,4,-13,2,-15)(-8,17,-20,5,11)(-9,-19,-17)(-10,7,-16,19)(-18,9)(3,13)(6,20,16)(8,10,18)
Loop annotated with half-edges
12^1_82 annotated with half-edges