$12^{1}_{111}$ - Minimal pinning sets
Pinning sets for 12^1_111 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_111 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 384
of which optimal: 2
of which minimal: 2
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: 2.25
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, 3, 4, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 2, 3, 11}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
15
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
0
382
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,3],[0,2,8,9],[0,6,5,5],[1,4,4,1],[1,4,9,2],[2,9,8,8],[3,7,7,9],[3,8,7,6]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[3,14,4,15],[19,10,20,11],[16,10,17,9],[1,12,2,13],[13,2,14,3],[4,12,5,11],[18,7,19,8],[17,7,18,6],[8,5,9,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(6,3,-7,-4)(4,15,-5,-16)(16,5,-17,-6)(7,14,-8,-15)(17,8,-18,-9)(9,2,-10,-3)(19,10,-20,-11)(1,12,-2,-13)(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,-13,-19,-11)(-2,9,-18,13)(-3,6,-17,-9)(-4,-16,-6)(-5,16)(-7,-15,4)(-8,17,5,15)(-10,19,-14,7,3)(-12,1)(-20,11)(2,12,20,10)(8,14,18)
Loop annotated with half-edges
12^1_111 annotated with half-edges