$12^{1}_{56}$ - Minimal pinning sets
Pinning sets for 12^1_56 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_56 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, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 5, 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, 3, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,5,0],[0,4,1,1],[1,3,6,7],[2,7,8,2],[4,8,9,9],[4,9,8,5],[5,7,9,6],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[13,2,14,3],[19,4,20,5],[1,12,2,13],[14,12,15,11],[5,18,6,19],[15,9,16,8],[17,10,18,11],[6,10,7,9],[16,7,17,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(17,2,-18,-3)(14,5,-15,-6)(6,13,-7,-14)(7,4,-8,-5)(15,8,-16,-9)(9,12,-10,-13)(19,10,-20,-11)(3,16,-4,-17)(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,-11)(-2,17,-4,7,13,-10,19)(-3,-17)(-5,14,-7)(-6,-14)(-8,15,5)(-9,-13,6,-15)(-12,9,-16,3,-18,1)(-20,11)(2,18)(4,16,8)(10,12,20)
Loop annotated with half-edges
12^1_56 annotated with half-edges