$12^{1}_{196}$ - Minimal pinning sets
Pinning sets for 12^1_196 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_196 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
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, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 5, 9, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,5],[0,5,6,7],[1,7,2,1],[2,8,3,2],[3,8,9,7],[3,6,9,4],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,18,11,17],[1,12,2,13],[7,18,8,19],[11,16,12,17],[2,5,3,6],[13,6,14,7],[4,15,5,16],[3,15,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (19,2,-20,-3)(10,3,-11,-4)(16,5,-17,-6)(18,9,-19,-10)(11,20,-12,-1)(1,12,-2,-13)(8,13,-9,-14)(14,7,-15,-8)(4,15,-5,-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,-13,8,-15,4,-11)(-2,19,9,13)(-3,10,-19)(-4,-16,-6,-18,-10)(-5,16)(-7,14,-9,18)(-8,-14)(-12,1)(-17,6)(-20,11,3)(2,12,20)(5,15,7,17)
Loop annotated with half-edges
12^1_196 annotated with half-edges