$12^{1}_{136}$ - Minimal pinning sets
Pinning sets for 12^1_136 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_136 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, 6, 9, 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, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,7,7,8],[0,4,4,0],[1,3,3,9],[1,9,6,6],[1,5,5,7],[2,6,8,2],[2,7,9,9],[4,8,8,5]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[3,6,4,7],[19,12,20,13],[8,2,9,1],[9,2,10,3],[16,5,17,6],[4,17,5,18],[13,18,14,19],[14,11,15,12],[10,15,11,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(14,3,-15,-4)(1,4,-2,-5)(12,9,-13,-10)(17,10,-18,-11)(8,13,-9,-14)(2,15,-3,-16)(11,16,-12,-17)(5,18,-6,-19)(19,6,-20,-7)
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,-5,-19,-7)(-2,-16,11,-18,5)(-3,14,-9,12,16)(-4,1,-8,-14)(-6,19)(-10,17,-12)(-11,-17)(-13,8,20,6,18,10)(-15,2,4)(-20,7)(3,15)(9,13)
Loop annotated with half-edges
12^1_136 annotated with half-edges