$12^{2}_{232}$ - Minimal pinning sets
Pinning sets for 12^2_232 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_232 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
C (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,7,3,3],[0,2,2,8],[0,8,9,5],[1,4,6,1],[1,5,7,7],[2,6,6,9],[3,9,9,4],[4,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[7,10,8,1],[6,20,7,11],[9,17,10,18],[8,17,9,16],[1,4,2,5],[11,5,12,6],[12,19,13,20],[18,13,19,14],[3,15,4,16],[2,15,3,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(18,3,-19,-4)(5,8,-6,-9)(15,20,-16,-11)(11,10,-12,-1)(1,12,-2,-13)(13,6,-14,-7)(7,14,-8,-15)(19,16,-20,-17)(4,17,-5,-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,-13,-7,-15,-11)(-2,9,-6,13)(-3,18,-5,-9)(-4,-18)(-8,5,17,-20,15)(-10,11,-16,19,3)(-12,1)(-14,7)(-17,4,-19)(2,12,10)(6,8,14)(16,20)
Multiloop annotated with half-edges
12^2_232 annotated with half-edges