$12^{1}_{307}$ - Minimal pinning sets
Pinning sets for 12^1_307 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_307 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 48
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: 2.86152
on average over minimal pinning sets: 2.14286
on average over optimal pinning sets: 2.14286
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 7, 8, 9}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
B (optimal)
• {1, 2, 3, 4, 6, 8, 9}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
2
0
0
2.14
8
0
0
9
2.53
9
0
0
16
2.82
10
0
0
14
3.04
11
0
0
6
3.21
12
0
0
1
3.33
Total
2
0
46
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 5, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,6,0],[1,7,2,1],[2,8,8,6],[3,5,7,3],[4,6,9,9],[5,9,9,5],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[8,1,9,2],[18,12,19,11],[6,15,7,16],[2,7,3,8],[12,3,13,4],[14,5,15,6],[13,5,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(1,10,-2,-11)(11,2,-12,-3)(19,4,-20,-5)(17,6,-18,-7)(8,15,-9,-16)(3,12,-4,-13)(16,13,-17,-14)(14,7,-15,-8)(5,18,-6,-19)
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,-3,-13,16,-9)(-2,11)(-4,19,-6,17,13)(-5,-19)(-7,14,-17)(-8,-16,-14)(-10,1)(-12,3)(-15,8)(-18,5,-20,9,15,7)(2,10,20,4,12)(6,18)
Loop annotated with half-edges
12^1_307 annotated with half-edges