$12^{1}_{76}$ - Minimal pinning sets
Pinning sets for 12^1_76 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_76 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 256
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: 3.0346
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
0
19
2.68
7
0
0
51
2.89
8
0
0
75
3.03
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
253
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,3],[0,2,7,8],[0,6,5,5],[1,4,4,1],[1,4,9,2],[2,9,8,3],[3,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[3,12,4,13],[19,8,20,9],[14,8,15,7],[1,10,2,11],[11,2,12,3],[4,10,5,9],[18,15,19,16],[6,17,7,18],[5,17,6,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(5,12,-6,-13)(17,6,-18,-7)(7,2,-8,-3)(19,8,-20,-9)(1,10,-2,-11)(16,13,-17,-14)(14,3,-15,-4)(4,15,-5,-16)(11,18,-12,-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,-19,-9)(-2,7,-18,11)(-3,14,-17,-7)(-4,-16,-14)(-5,-13,16)(-6,17,13)(-8,19,-12,5,15,3)(-10,1)(-15,4)(-20,9)(2,10,20,8)(6,12,18)
Loop annotated with half-edges
12^1_76 annotated with half-edges