$12^{1}_{192}$ - Minimal pinning sets
Pinning sets for 12^1_192 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_192 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
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, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,8,8],[0,9,6,5],[1,4,2,1],[2,4,7,7],[3,6,6,9],[3,9,9,3],[4,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[19,16,20,17],[6,13,7,14],[1,11,2,10],[18,10,19,9],[2,15,3,16],[14,3,15,4],[12,5,13,6],[11,5,12,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(15,2,-16,-3)(4,13,-5,-14)(5,16,-6,-17)(1,8,-2,-9)(17,10,-18,-11)(14,11,-15,-12)(12,3,-13,-4)(9,18,-10,-19)(19,6,-20,-7)
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,-9,-19,-7)(-2,15,11,-18,9)(-3,12,-15)(-4,-14,-12)(-5,-17,-11,14)(-6,19,-10,17)(-8,1)(-13,4)(-16,5,13,3)(-20,7)(2,8,20,6,16)(10,18)
Loop annotated with half-edges
12^1_192 annotated with half-edges