$12^{1}_{314}$ - Minimal pinning sets
Pinning sets for 12^1_314 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_314 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 48
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.86152
on average over minimal pinning sets: 2.14286
on average over optimal pinning sets: 2.14286
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 7, 9, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
B (optimal)
• {1, 2, 3, 6, 8, 9, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
2
0
0
2.14
8
0
0
9
2.53
9
0
0
16
2.82
10
0
0
14
3.04
11
0
0
6
3.21
12
0
0
1
3.33
Total
2
0
46
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 7, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,6],[1,7,5,5],[2,4,4,7],[3,8,8,3],[4,9,9,5],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[19,14,20,15],[16,14,17,13],[1,13,2,12],[18,7,19,8],[17,7,18,6],[2,11,3,12],[8,5,9,6],[10,3,11,4],[4,9,5,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(5,2,-6,-3)(3,14,-4,-15)(15,4,-16,-5)(1,6,-2,-7)(19,8,-20,-9)(9,18,-10,-19)(10,13,-11,-14)(16,11,-17,-12)(12,17,-13,-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,-7)(-2,5,-16,-12,-18,9,-20,7)(-3,-15,-5)(-4,15)(-6,1,-8,19,-10,-14,3)(-9,-19)(-11,16,4,14)(-13,10,18)(-17,12)(2,6)(8,20)(11,13,17)
Loop annotated with half-edges
12^1_314 annotated with half-edges