$12^{1}_{270}$ - Minimal pinning sets
Pinning sets for 12^1_270 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_270 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 88
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91951
on average over minimal pinning sets: 2.29365
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, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 4, 6, 7, 9, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
b (minimal)
• {1, 2, 4, 5, 6, 9, 11}
7
[2, 2, 2, 2, 2, 3, 4]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
2
6
2.46
8
0
0
22
2.74
9
0
0
29
2.97
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
2
85
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,6,7,7],[1,8,6,5],[2,4,6,2],[3,5,4,8],[3,9,9,3],[4,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,8,11,7],[1,17,2,16],[5,18,6,19],[11,6,12,7],[17,12,18,13],[2,15,3,16],[4,13,5,14],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(8,3,-9,-4)(4,7,-5,-8)(16,5,-17,-6)(14,9,-15,-10)(1,10,-2,-11)(19,12,-20,-13)(13,18,-14,-19)(2,15,-3,-16)(6,17,-7,-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)(-2,-16,-6,-18,13,-20,11)(-3,8,-5,16)(-4,-8)(-7,4,-9,14,18)(-10,1,-12,19,-14)(-13,-19)(-15,2,10)(-17,6)(3,15,9)(5,7,17)(12,20)
Loop annotated with half-edges
12^1_270 annotated with half-edges