$12^{1}_{291}$ - Minimal pinning sets
Pinning sets for 12^1_291 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_291 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 120
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91853
on average over minimal pinning sets: 2.25
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, 4, 6, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 5, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 3, 4, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
D (optimal)
• {1, 2, 3, 4, 8, 11}
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
4
0
0
2.25
7
0
0
18
2.6
8
0
0
34
2.85
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
4
0
116
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,7,8,8],[0,9,9,5],[1,4,6,1],[2,5,7,7],[2,6,6,3],[3,9,9,3],[4,8,8,4]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,15,7,16],[16,19,17,20],[4,11,5,12],[1,9,2,8],[14,7,15,8],[18,13,19,14],[17,13,18,12],[10,3,11,4],[9,3,10,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(8,1,-9,-2)(18,3,-19,-4)(12,5,-13,-6)(6,19,-7,-20)(2,9,-3,-10)(10,15,-11,-16)(16,11,-17,-12)(4,13,-5,-14)(14,17,-15,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-10,-16,-12,-6,-20,-8)(-3,18,-15,10)(-4,-14,-18)(-5,12,-17,14)(-7,20)(-9,2)(-11,16)(-13,4,-19,6)(1,7,19,3,9)(5,13)(11,15,17)
Loop annotated with half-edges
12^1_291 annotated with half-edges