$12^{1}_{319}$ - Minimal pinning sets
Pinning sets for 12^1_319 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_319 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, 2, 3, 4, 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, 3, 6, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,5,5,0],[1,6,6,4],[1,3,7,7],[2,8,9,2],[3,9,9,3],[4,8,8,4],[5,7,7,9],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,19,13,20],[10,1,11,2],[5,18,6,19],[13,6,14,7],[2,9,3,10],[17,4,18,5],[14,8,15,7],[15,8,16,9],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(10,3,-11,-4)(4,11,-5,-12)(12,5,-13,-6)(19,6,-20,-7)(17,8,-18,-9)(20,13,-1,-14)(2,15,-3,-16)(9,16,-10,-17)(7,18,-8,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14)(-2,-16,9,-18,7,-20,-14)(-3,10,16)(-4,-12,-6,19,-8,17,-10)(-5,12)(-7,-19)(-9,-17)(-11,4)(-13,20,6)(-15,2)(1,13,5,11,3,15)(8,18)
Loop annotated with half-edges
12^1_319 annotated with half-edges