$12^{1}_{316}$ - Minimal pinning sets
Pinning sets for 12^1_316 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_316 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 32
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.80821
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 6, 7, 8, 9}
7
[2, 2, 2, 2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
1
0
0
2.0
8
0
0
5
2.4
9
0
0
10
2.71
10
0
0
10
2.96
11
0
0
5
3.16
12
0
0
1
3.33
Total
1
0
31
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,4,0],[1,5,6,1],[2,6,7,2],[3,7,7,8],[3,9,9,4],[4,8,5,5],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[19,6,20,7],[8,2,9,1],[5,18,6,19],[2,10,3,9],[4,13,5,14],[17,10,18,11],[3,15,4,14],[15,12,16,13],[11,16,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,3,-17,-4)(6,19,-7,-20)(14,7,-15,-8)(12,9,-13,-10)(1,10,-2,-11)(11,20,-12,-1)(8,13,-9,-14)(2,15,-3,-16)(4,17,-5,-18)(18,5,-19,-6)
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,-4,-18,-6,-20,11)(-3,16)(-5,18)(-7,14,-9,12,20)(-8,-14)(-10,1,-12)(-13,8,-15,2,10)(-17,4)(-19,6)(3,15,7,19,5,17)(9,13)
Loop annotated with half-edges
12^1_316 annotated with half-edges