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