$12^{1}_{20}$ - Minimal pinning sets
Pinning sets for 12^1_20 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_20 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, 2, 5, 7}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 5, 7}
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, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,6,3],[0,2,1,0],[1,6,6,7],[1,7,8,2],[2,9,4,4],[4,9,8,5],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,14,17,13],[19,2,20,3],[14,1,15,2],[17,9,18,8],[12,3,13,4],[18,9,19,10],[7,4,8,5],[11,6,12,7],[10,6,11,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(4,1,-5,-2)(2,7,-3,-8)(8,3,-9,-4)(5,14,-6,-15)(15,6,-16,-7)(10,13,-11,-14)(17,12,-18,-13)(11,18,-12,-19)(16,19,-17,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-9)(-2,-8,-4)(-3,8)(-5,-15,-7,2)(-6,15)(-10,-14,5,1)(-11,-19,16,6,14)(-12,17,19)(-13,10,20,-17)(-16,-20,9,3,7)(-18,11,13)(12,18)
Loop annotated with half-edges
12^1_20 annotated with half-edges