$12^{1}_{215}$ - Minimal pinning sets
Pinning sets for 12^1_215 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_215 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
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, 3, 4, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 3, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
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: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,6,6],[0,7,8,8],[1,8,9,1],[2,9,3,3],[4,9,9,8],[4,7,5,4],[5,7,7,6]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[15,4,16,5],[16,19,17,20],[6,17,7,18],[1,10,2,11],[3,14,4,15],[18,7,19,8],[9,12,10,13],[2,12,3,11],[13,8,14,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(16,5,-17,-6)(11,6,-12,-7)(19,8,-20,-9)(9,20,-10,-1)(7,10,-8,-11)(1,12,-2,-13)(4,15,-5,-16)(14,17,-15,-18)(3,18,-4,-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,-3,-19,-9)(-2,13)(-4,-16,-6,11,-8,19)(-5,16)(-7,-11)(-10,7,-12,1)(-14,-18,3)(-15,4,18)(-17,14,2,12,6)(-20,9)(5,15,17)(8,10,20)
Loop annotated with half-edges
12^1_215 annotated with half-edges