$12^{2}_{321}$ - Minimal pinning sets
Pinning sets for 12^2_321 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_321 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 344
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04482
on average over minimal pinning sets: 2.50417
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 3, 5, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.58
6
0
2
34
2.79
7
0
0
78
2.95
8
0
0
99
3.07
9
0
0
76
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
3
340
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,6,7],[0,7,7,4],[0,3,8,1],[1,8,9,6],[1,5,2,2],[2,9,3,3],[4,9,9,5],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,7,12,8],[17,9,18,10],[5,19,6,20],[1,6,2,7],[12,15,13,16],[8,16,9,17],[18,4,19,5],[2,14,3,15],[13,3,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-11,-2)(15,2,-16,-3)(9,4,-10,-5)(18,5,-19,-6)(14,7,-15,-8)(10,11,-1,-12)(19,12,-20,-13)(6,13,-7,-14)(3,16,-4,-17)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,12)(-2,15,7,13,-20)(-3,-17,8,-15)(-4,9,17)(-5,18,-9)(-6,-14,-8,-18)(-7,14)(-10,-12,19,5)(-11,10,4,16,2)(-13,6,-19)(-16,3)(1,11)
Multiloop annotated with half-edges
12^2_321 annotated with half-edges