$12^{1}_{226}$ - Minimal pinning sets
Pinning sets for 12^1_226 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_226 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 5, 6, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,7,7,8],[0,8,8,4],[0,3,8,9],[1,9,9,1],[1,9,7,7],[2,6,6,2],[2,4,3,3],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,7,15,8],[10,19,11,20],[3,12,4,13],[1,4,2,5],[6,15,7,16],[8,17,9,18],[18,9,19,10],[11,2,12,3],[5,17,6,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(19,2,-20,-3)(12,5,-13,-6)(16,7,-17,-8)(1,10,-2,-11)(18,11,-19,-12)(4,13,-5,-14)(14,3,-15,-4)(6,15,-7,-16)(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,-11,18,-9)(-2,19,11)(-3,14,-5,12,-19)(-4,-14)(-6,-16,-8,-18,-12)(-7,16)(-10,1)(-13,4,-15,6)(-17,8)(-20,9,17,7,15,3)(2,10,20)(5,13)
Loop annotated with half-edges
12^1_226 annotated with half-edges