$12^{1}_{292}$ - Minimal pinning sets
Pinning sets for 12^1_292 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_292 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 112
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: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 4, 5, 7, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
C (optimal)
• {1, 3, 4, 7, 8, 9}
6
[2, 2, 2, 2, 2, 4]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,5,0],[1,6,7,1],[2,7,8,5],[2,4,9,6],[3,5,9,9],[3,8,8,4],[4,7,7,9],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,19,13,20],[10,1,11,2],[18,13,19,14],[2,5,3,6],[6,9,7,10],[14,7,15,8],[4,17,5,18],[3,17,4,16],[8,15,9,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,3,-1,-4)(12,5,-13,-6)(4,7,-5,-8)(18,9,-19,-10)(16,11,-17,-12)(6,13,-7,-14)(14,1,-15,-2)(2,15,-3,-16)(10,17,-11,-18)(8,19,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-7,4)(-2,-16,-12,-6,-14)(-3,20,-9,18,-11,16)(-4,-8,-20)(-5,12,-17,10,-19,8)(-10,-18)(-13,6)(-15,2)(1,3,15)(5,7,13)(9,19)(11,17)
Loop annotated with half-edges
12^1_292 annotated with half-edges