$12^{3}_{11}$ - Minimal pinning sets
Pinning sets for 12^3_11 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_11 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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, 12}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,8],[0,8,8,9],[0,6,5,5],[1,4,4,1],[1,4,9,2],[2,9,9,8],[2,7,3,3],[3,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,3,10,4],[13,7,14,8],[15,20,16,17],[1,12,2,11],[2,10,3,11],[4,12,5,13],[6,18,7,19],[14,18,15,17],[19,5,20,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(14,5,-15,-6)(10,7,-11,-8)(4,15,-5,-16)(16,3,-17,-4)(6,17,-7,-18)(2,13,-3,-14)(20,11,-13,-12)(12,19,-9,-20)(8,9,-1,-10)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7,10)(-2,-14,-6,-18)(-3,16,-5,14)(-4,-16)(-8,-10)(-9,8,-11,20)(-12,-20)(-13,2,-19,12)(-15,4,-17,6)(1,9,19)(3,13,11,7,17)(5,15)
Multiloop annotated with half-edges
12^3_11 annotated with half-edges