Polytopes Table

This table contains Hilbert bases (HB) and Markov bases (MB) of affine semigroups obtained as images of maximal rank valuations on the coordinate rings of three different classes of
algebraic varieties. The number of elements in the Hilbert and Markov bases are
upper bounds on the number of generators and relations needed to present the
associated coordinate ring. Information about each variety appears below, more
about the techniques used to produce these semigroups can be found in the papers
https://arxiv.org/abs/1403.3990 and https://arxiv.org/abs/1103.2484. All
computations are carried out in 4ti2 http://www.4ti2.de/.

Graphs and Character Varieties

dumbbell.svg
$$SL_2(\mathbb{C})$$ HB: 3 | MB: 0
$$SL_3(\mathbb{C})$$ HB: 10 | MB: 2
$$SL_4(\mathbb{C})$$ HB: 89 | MB: 2240
$$SL_5(\mathbb{C})$$ HB: 12955
pizza.svg
$$SL_2(\mathbb{C})$$ HB: 3 | MB: 0
$$SL_3(\mathbb{C})$$ HB: 10 | MB: 2
$$SL_4(\mathbb{C})$$ HB: 44 | MB: 360
$$SL_5(\mathbb{C})$$ HB: 1561
k4
$$SL_2(\mathbb{C})$$ HB: 7 | MB: 1
$$SL_3(\mathbb{C})$$ HB: 65 | MB: 1029
cater3
$$SL_2(\mathbb{C})$$ HB: 7 | MB: 1
$$SL_3(\mathbb{C})$$ HB: 341 | MB: 49583
dumbg3
$$SL_2(\mathbb{C})$$ HB: 8 | MB: 2
$$SL_3(\mathbb{C})$$ HB: 328 | MB: 64923
pend4
$$SL_2(\mathbb{C})$$ HB: 8 | MB: 2
$$SL_3(\mathbb{C})$$ HB: 213 | MB: 16563
box
$$SL_2(\mathbb{C})$$ HB: 8 | MB: 2
$$SL_3(\mathbb{C})$$ HB: 65 | MB: 1029
6V1.svg
$$SL_2(\mathbb{C})$$ HB: 23 | MB: 75
6V2.svg
$$SL_2(\mathbb{C})$$ HB: 22 | MB: 67
6V3correct.svg
$$SL_2(\mathbb{C})$$ HB: 21 | MB:60
6V4.svg
$$SL_2(\mathbb{C})$$ HB: 20 | MB: 55
6V5.svg
$$SL_2(\mathbb{C})$$ HB: 15 | MB: 20
6V6.svg
$$SL_2(\mathbb{C})$$ HB: 20 | MB: 45
6V7.svg
$$SL_2(\mathbb{C})$$ HB: 20 | MB: 44
6V8.svg
$$SL_2(\mathbb{C})$$ HB: 22 | MB: 63
6V9.svg
$$SL_2(\mathbb{C})$$ HB: 22 | MB: 63
6V10.svg
$$SL_2(\mathbb{C})$$ HB: 18 | MB: 36
6V11.svg
$$SL_2(\mathbb{C})$$ HB: 18 | MB: 36
6V12.svg
$$SL_2(\mathbb{C})$$ HB: 17 | MB: 26
6V13.svg
$$SL_2(\mathbb{C})$$ HB: 22 | MB: 63
6V14.svg
$$SL_2(\mathbb{C})$$ HB: 22 | MB: 63
6V15.svg
$$SL_2(\mathbb{C})$$ HB: 22 | MB: 63
6V16.svg
$$SL_2(\mathbb{C})$$ HB: 22 | MB: 63
6V17.svg
$$SL_2(\mathbb{C})$$ HB: 15 | MB: 24

Trees and Configurations of full principal Flags

trinion
$$SL_3(\mathbb{C})$$ HB: 8 | MB: 1
$$SL_4(\mathbb{C})$$ HB: 18 | MB: 15
$$SL_5(\mathbb{C})$$ HB: 45 | MB: 234
$$SL_6(\mathbb{C})$$ HB: 166 | MB: 7039
$$SL_7(\mathbb{C})$$ HB: 1369
$$SL_8(\mathbb{C})$$ HB: 39219
T1.svg
$$SL_2(\mathbb{C})$$ HB: 6 | MB: 1
$$SL_3(\mathbb{C})$$ HB: 22 | MB: 35
$$SL_4(\mathbb{C})$$ HB: 103 | MB: 2194
$$SL_5(\mathbb{C})$$ HB: 1515
T2.svg
$$SL_2(\mathbb{C})$$ HB: 10 | MB: 5
$$SL_3(\mathbb{C})$$ HB: 52 | MB: 364
$$SL_4(\mathbb{C})$$ HB: 754
T3.svg
$$SL_2(\mathbb{C})$$ HB: 15 | MB: 15
$$SL_3(\mathbb{C})$$ HB: 114 | MB: 2562
T4.svg
$$SL_2(\mathbb{C})$$ HB: 15 | MB: 15
$$SL_3(\mathbb{C})$$ HB: 114 | MB: 2530
T5.svg
$$SL_2(\mathbb{C})$$ HB: 21 | MB: 35
$$SL_3(\mathbb{C})$$ HB: 240
T6.svg
$$SL_2(\mathbb{C})$$ HB: 21 | MB: 35
$$SL_3(\mathbb{C})$$ HB: 240
8.svg
$$SL_2(\mathbb{C})$$ HB: 28 | MB: 70
$$SL_3(\mathbb{C})$$ HB: 494

Trees and Configurations of full Flags

btrinion.svg
$$SL_2(\mathbb{C})$$ HB: 1 | MB: 0
$$SL_3(\mathbb{C})$$ HB:2 | MB:0
$$SL_4(\mathbb{C})$$ HB 12 | MB: 28
$$SL_5(\mathbb{C})$$ HB: 22 | MB: 89
$$SL_6(\mathbb{C})$$ HB:4464
$$SL_7(\mathbb{C})$$ HB: 43177
BT1.svg
$$SL_2(\mathbb{C})$$ HB: 2 | MB: 0
$$SL_3(\mathbb{C})$$ HB: 8 | MB: 3
$$SL_4(\mathbb{C})$$ HB: 142 | MB: 7388
BT2.svg
$$SL_2(\mathbb{C})$$ HB: 6 | MB: 5
$$SL_3(\mathbb{C})$$ HB: 67 | MB: 1465
BT3.svg
$$SL_2(\mathbb{C})$$ HB: 5 | MB: 1
$$SL_3(\mathbb{C})$$ HB: 463
BT4.svg
$$SL_2(\mathbb{C})$$ HB: 6 | MB: 2
$$SL_3(\mathbb{C})$$ HB:1061
bt5
$$SL_2(\mathbb{C})$$ HB: 36 | MB: 406
$$SL_3(\mathbb{C})$$ HB: 16646
bt6.svg
$$SL_2(\mathbb{C})$$ HB: 36 | MB: 406

Configurations in projective space

Graph[].svg
$$SL_3(\mathbb{C})$$ HB: 1 | MB: 0
Graph[(0,1)].svg
$$SL_3(\mathbb{C})$$ HB: 1 | MB: 0
$$SL_4(\mathbb{C})$$ HB: 1 | MB: 0
Graph[(0,1),(1,2)].svg
$$SL_3(\mathbb{C})$$ HB: 6 | MB: 5
$$SL_4(\mathbb{C})$$ HB: 1 | MB: 0
$$SL_5(\mathbb{C})$$ HB: 1 | MB: 0
Graph[(0,1),(1,2),(2,3)].svg
$$SL_3(\mathbb{C})$$ HB: 7 | MB: 2
$$SL_4(\mathbb{C})$$ HB: 6 | MB: 2
Graph[(0,1),(1,2),(1,3)].svg
$$SL_3(\mathbb{C})$$ HB: 7 | MB: 2
$$SL_4(\mathbb{C})$$ HB: 5 | MB: 1
Graph[(0,1),(1,2),(2,3),(3,4)].svg
$$SL_3(\mathbb{C})$$ HB: 273 | MB: 32449
Graph[(0,1),(1,2),(2,3),(2,4)].svg
$$SL_3(\mathbb{C})$$ HB: 282 | MB: 34828
Graph[(0,1),(1,2),(2,3),(3,4),(4,5)].svg
$$SL_3(\mathbb{C})$$ HB: 2327 | MB:
[(0,1),(1,2),(2,3),(3,4),(4,5),(5,6)]
$$SL_3(\mathbb{C})$$ HB: 443 | MB:
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