$12^{1}_{346}$ - Minimal pinning sets
Pinning sets for 12^1_346 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_346 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 536
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.18967
on average over minimal pinning sets: 2.83214
on average over optimal pinning sets: 3.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 8, 9}
4
[2, 3, 3, 4]
3.00
a (minimal)
• {1, 2, 4, 5, 9}
5
[2, 3, 3, 3, 3]
2.80
b (minimal)
• {1, 2, 3, 5, 9}
5
[2, 3, 3, 3, 3]
2.80
c (minimal)
• {1, 2, 3, 9, 10}
5
[2, 3, 3, 3, 3]
2.80
d (minimal)
• {1, 3, 5, 7, 9}
5
[2, 3, 3, 3, 3]
2.80
e (minimal)
• {1, 3, 7, 9, 10}
5
[2, 3, 3, 3, 3]
2.80
f (minimal)
• {1, 3, 7, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
g (minimal)
• {1, 2, 4, 5, 7, 10, 11}
7
[2, 3, 3, 3, 3, 3, 3]
2.86
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
3.0
5
0
6
8
2.97
6
0
0
60
3.06
7
0
1
127
3.14
8
0
0
157
3.21
9
0
0
115
3.26
10
0
0
49
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
1
7
528
Other information about this loop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,8,4],[0,3,8,1],[1,9,7,6],[1,5,7,2],[2,6,5,9],[3,9,9,4],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,9,17,10],[6,19,7,20],[7,14,8,15],[1,8,2,9],[17,4,18,5],[10,5,11,6],[11,18,12,19],[13,2,14,3],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(15,2,-16,-3)(8,3,-9,-4)(19,6,-20,-7)(14,7,-15,-8)(1,10,-2,-11)(5,12,-6,-13)(18,13,-19,-14)(9,16,-10,-17)(4,17,-5,-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,-11)(-2,15,7,-20,11)(-3,8,-15)(-4,-18,-14,-8)(-5,-13,18)(-6,19,13)(-7,14,-19)(-9,-17,4)(-10,1,-12,5,17)(-16,9,3)(2,10,16)(6,12,20)
Loop annotated with half-edges
12^1_346 annotated with half-edges