$12^{1}_{323}$ - Minimal pinning sets
Pinning sets for 12^1_323 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_323 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 256
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0346
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 7, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 7, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
0
19
2.68
7
0
0
51
2.89
8
0
0
75
3.03
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
253
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,4],[0,3,5,5],[1,4,4,8],[1,8,7,2],[3,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[3,12,4,13],[4,19,5,20],[14,5,15,6],[1,6,2,7],[7,2,8,3],[18,11,19,12],[15,11,16,10],[8,17,9,18],[16,9,17,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(12,3,-13,-4)(19,4,-20,-5)(14,7,-15,-8)(8,13,-9,-14)(16,9,-17,-10)(1,10,-2,-11)(6,15,-7,-16)(2,17,-3,-18)(11,18,-12,-19)
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,-11,-19,-5)(-2,-18,11)(-3,12,18)(-4,19,-12)(-6,-16,-10,1)(-7,14,-9,16)(-8,-14)(-13,8,-15,6,20,4)(-17,2,10)(-20,5)(3,17,9,13)(7,15)
Loop annotated with half-edges
12^1_323 annotated with half-edges