$12^{4}_{7}$ - Minimal pinning sets
Pinning sets for 12^4_7 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_7 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, 2, 4, 7, 12}
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, 4, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,6,7,0],[0,7,7,4],[1,3,8,1],[1,8,8,6],[2,5,9,9],[2,9,3,3],[4,9,5,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[4,12,1,5],[5,13,6,16],[11,3,12,4],[1,8,2,7],[13,7,14,6],[15,20,16,17],[10,19,11,20],[2,8,3,9],[14,18,15,17],[18,9,19,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(11,18,-12,-19)(12,3,-5,-4)(4,5,-1,-6)(13,6,-14,-7)(2,9,-3,-10)(17,10,-18,-11)(7,14,-8,-15)(20,15,-17,-16)(16,19,-13,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,14,6)(-2,-10,17,15,-8)(-3,12,18,10)(-4,-6,13,19,-12)(-5,4)(-7,-15,20,-13)(-9,2)(-11,-19,16,-17)(-14,7)(-16,-20)(-18,11)(1,5,3,9)
Multiloop annotated with half-edges
12^4_7 annotated with half-edges