$12^{2}_{53}$ - Minimal pinning sets
Pinning sets for 12^2_53 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_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)
• {2, 5, 6, 9, 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, 3, 3, 3, 3, 4, 6, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,3],[0,2,8,8],[0,5,1,1],[1,4,9,9],[2,9,9,7],[2,6,8,8],[3,7,7,3],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,20,3,9],[13,7,14,8],[4,14,5,15],[1,10,2,9],[10,19,11,20],[12,17,13,18],[6,16,7,17],[5,16,6,15],[18,11,19,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,5,-15,-6)(1,6,-2,-7)(19,12,-20,-13)(2,15,-3,-16)(16,3,-17,-4)(4,17,-5,-18)(13,18,-14,-19)(11,20,-12,-9)(8,9,-1,-10)(10,7,-11,-8)
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,-7,10)(-2,-16,-4,-18,13,-20,11,7)(-3,16)(-5,14,18)(-6,1,9,-12,19,-14)(-8,-10)(-9,8,-11)(-13,-19)(-15,2,6)(-17,4)(3,15,5,17)(12,20)
Multiloop annotated with half-edges
12^2_53 annotated with half-edges