$12^{1}_{115}$ - Minimal pinning sets
Pinning sets for 12^1_115 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_115 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, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 5, 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, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,7,8],[0,5,1,1],[1,4,8,6],[2,5,9,2],[3,9,9,8],[3,7,9,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,15,3,16],[8,19,9,20],[4,9,5,10],[1,17,2,16],[17,14,18,15],[18,7,19,8],[5,12,6,13],[10,13,11,14],[11,6,12,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,4,-10,-5)(16,5,-17,-6)(2,7,-3,-8)(3,10,-4,-11)(8,11,-9,-12)(15,12,-16,-13)(20,13,-1,-14)(14,19,-15,-20)(6,17,-7,-18)(1,18,-2,-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,-19,14)(-2,-8,-12,15,19)(-3,-11,8)(-4,9,11)(-5,16,12,-9)(-6,-18,1,13,-16)(-7,2,18)(-10,3,7,17,5)(-13,20,-15)(-14,-20)(-17,6)(4,10)
Loop annotated with half-edges
12^1_115 annotated with half-edges