$12^{2}_{94}$ - Minimal pinning sets
Pinning sets for 12^2_94
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_94
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 256
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: 3.0346
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{2, 4, 6, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
•
{2, 4, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
•
{2, 4, 5, 7, 11}
5
[2, 2, 2, 3, 3]
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
2.4
6
0
0
19
2.68
7
0
0
51
2.89
8
0
0
75
3.03
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
253
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,3],[0,2,7,8],[0,5,1,1],[1,4,6,6],[2,5,5,9],[2,9,8,3],[3,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[9,17,10,18],[4,17,5,16],[1,12,2,11],[12,19,13,20],[18,13,19,14],[8,5,9,6],[15,7,16,8],[14,7,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,3,-7,-4)(4,15,-5,-16)(16,5,-17,-6)(1,8,-2,-9)(19,14,-20,-15)(17,2,-18,-3)(7,18,-8,-19)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,17,5,15,-20,13,9)(-3,6,-17)(-4,-16,-6)(-5,16)(-7,-19,-15,4)(-8,1,11,-14,19)(-10,-12)(-11,10,-13)(-18,7,3)(2,8,18)(14,20)
Multiloop annotated with half-edges
12^2_94 annotated with half-edges