$12^{1}_{126}$ - Minimal pinning sets
Pinning sets for 12^1_126 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_126 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, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 3, 4, 8, 9, 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],[0,4,5,6],[0,7,8,8],[0,8,8,9],[0,5,5,1],[1,4,4,6],[1,5,9,7],[2,6,9,9],[2,3,3,2],[3,7,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,7,11,8],[19,14,20,15],[12,18,13,17],[1,9,2,8],[2,9,3,10],[3,6,4,7],[15,4,16,5],[13,18,14,19],[5,16,6,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(20,3,-1,-4)(7,4,-8,-5)(5,18,-6,-19)(19,6,-20,-7)(16,11,-17,-12)(2,13,-3,-14)(14,9,-15,-10)(10,15,-11,-16)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-17,8,4)(-2,-14,-10,-16,-12)(-3,20,6,18,-9,14)(-4,7,-20)(-5,-19,-7)(-6,19)(-8,-18,5)(-11,16)(-13,2)(-15,10)(1,3,13)(9,17,11,15)
Loop annotated with half-edges
12^1_126 annotated with half-edges