$12^{1}_{243}$ - Minimal pinning sets
Pinning sets for 12^1_243 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_243 Pinning data
Pinning number of this loop: 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, 2, 4, 5, 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 loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,7,8,8],[0,8,8,9],[0,9,5,5],[1,4,4,9],[1,7,7,1],[2,6,6,9],[2,3,3,2],[3,7,5,4]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,3,11,4],[6,19,7,20],[8,17,9,18],[1,14,2,15],[15,2,16,3],[11,5,12,4],[12,5,13,6],[18,7,19,8],[13,16,14,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(8,1,-9,-2)(2,9,-3,-10)(10,3,-11,-4)(16,5,-17,-6)(18,7,-19,-8)(19,12,-20,-13)(11,14,-12,-15)(6,15,-7,-16)(4,17,-5,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-19,-13)(-2,-10,-4,-18,-8)(-3,10)(-5,16,-7,18)(-6,-16)(-9,2)(-11,-15,6,-17,4)(-12,19,7,15)(-14,11,3,9,1)(-20,13)(5,17)(12,14,20)
Loop annotated with half-edges
12^1_243 annotated with half-edges