$12^{1}_{244}$ - Minimal pinning sets
Pinning sets for 12^1_244 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_244 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 5, 11}
5
[2, 2, 2, 2, 2]
2.00
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
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,6,7,7],[0,8,8,9],[0,9,5,5],[1,4,4,9],[1,9,2,1],[2,8,8,2],[3,7,7,3],[3,6,5,4]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[3,14,4,15],[19,12,20,13],[16,10,17,9],[1,6,2,7],[7,2,8,3],[13,4,14,5],[11,18,12,19],[10,18,11,17],[5,8,6,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(11,2,-12,-3)(19,4,-20,-5)(3,6,-4,-7)(7,14,-8,-15)(17,10,-18,-11)(1,12,-2,-13)(15,8,-16,-9)(9,16,-10,-17)(13,18,-14,-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,-19,-5)(-2,11,-18,13)(-3,-7,-15,-9,-17,-11)(-4,19,-14,7)(-6,3,-12,1)(-8,15)(-10,17)(-16,9)(-20,5)(2,12)(4,6,20)(8,14,18,10,16)
Loop annotated with half-edges
12^1_244 annotated with half-edges