$12^{1}_{312}$ - Minimal pinning sets
Pinning sets for 12^1_312 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_312 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 64
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: 2.85421
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, 4, 5, 6, 9}
6
[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
6
1
0
0
2.0
7
0
0
6
2.38
8
0
0
15
2.67
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
1
0
63
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 4, 4, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,5,6,6],[0,6,7,7],[1,5,5,1],[2,4,4,8],[2,9,3,2],[3,9,8,3],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[13,8,14,9],[1,7,2,6],[18,11,19,12],[12,17,13,18],[7,14,8,15],[2,5,3,6],[3,16,4,17],[15,4,16,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(19,2,-20,-3)(3,18,-4,-19)(10,5,-11,-6)(17,6,-18,-7)(15,8,-16,-9)(4,11,-5,-12)(20,13,-1,-14)(9,14,-10,-15)(7,16,-8,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-5,10,14)(-2,19,-4,-12)(-3,-19)(-6,17,-8,15,-10)(-7,-17)(-9,-15)(-11,4,18,6)(-13,20,2)(-14,9,-16,7,-18,3,-20)(1,13)(5,11)(8,16)
Loop annotated with half-edges
12^1_312 annotated with half-edges