$12^{4}_{4}$ - Minimal pinning sets
Pinning sets for 12^4_4 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_4 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, 4, 8, 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, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,6,4,4],[1,3,3,7],[2,8,9,2],[3,9,7,7],[4,6,6,8],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[4,12,1,5],[5,3,6,4],[6,11,7,12],[1,13,2,16],[2,15,3,16],[10,7,11,8],[13,17,14,20],[14,19,15,20],[8,19,9,18],[9,17,10,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(19,16,-20,-17)(15,20,-16,-13)(2,13,-3,-14)(4,5,-1,-6)(6,3,-7,-4)(12,17,-9,-18)(8,9,-5,-10)(10,7,-11,-8)(18,11,-19,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,6)(-2,-14)(-4,-6)(-5,4,-7,10)(-8,-10)(-9,8,-11,18)(-12,-18)(-13,2,-15)(-16,19,11,7,3,13)(-17,12,-19)(-20,15,1,5,9,17)(16,20)
Multiloop annotated with half-edges
12^4_4 annotated with half-edges