$12^{3}_{53}$ - Minimal pinning sets
Pinning sets for 12^3_53 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_53 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
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: 2.90623
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, 5, 7, 11}
5
[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
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,6,7,7],[1,7,8,8],[1,8,2,2],[2,9,9,3],[3,9,4,3],[4,9,5,4],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[12,16,1,13],[13,11,14,12],[15,7,16,8],[1,17,2,20],[3,10,4,11],[14,9,15,8],[6,17,7,18],[2,19,3,20],[9,4,10,5],[18,5,19,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(16,3,-13,-4)(18,5,-19,-6)(6,17,-7,-18)(1,8,-2,-9)(20,9,-17,-10)(10,19,-11,-20)(12,13,-1,-14)(14,11,-15,-12)(4,15,-5,-16)
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,-9,20,-11,14)(-2,7,17,9)(-3,16,-5,18,-7)(-4,-16)(-6,-18)(-8,1,13,3)(-10,-20)(-12,-14)(-13,12,-15,4)(-17,6,-19,10)(2,8)(5,15,11,19)
Multiloop annotated with half-edges
12^3_53 annotated with half-edges