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Flat knot 6.1501

Min(phi) over symmetries of the knot is: [-2,0,0,1,1,1,1,1,2,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1501', '7.11373', '7.44314']
Arrow polynomial of the knot is: -6*K1**2 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.689', '6.691', '6.752', '6.754', '6.1106', '6.1116', '6.1126', '6.1335', '6.1379', '6.1386', '6.1409', '6.1415', '6.1417', '6.1418', '6.1421', '6.1422', '6.1428', '6.1431', '6.1432', '6.1435', '6.1443', '6.1445', '6.1446', '6.1447', '6.1454', '6.1455', '6.1460', '6.1462', '6.1464', '6.1466', '6.1472', '6.1474', '6.1475', '6.1501', '6.1516', '6.1518', '6.1566', '6.1570', '6.1590', '6.1599', '6.1602', '6.1603', '6.1604', '6.1605', '6.1614', '6.1615', '6.1625', '6.1628', '6.1730', '6.1780', '6.1883', '6.1885', '6.1888', '6.1890', '6.1941', '6.1943', '6.1945', '6.1948', '6.1961', '6.1963', '6.1966', '6.1967', '6.1971']
Outer characteristic polynomial of the knot is: t^6+17t^4+7t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1501', '7.44314']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 784*K1**4 - 512*K1**2*K2**2 + 1672*K1**2*K2 - 16*K1**2*K3**2 - 880*K1**2 + 568*K1*K2*K3 + 32*K1*K3*K4 - 56*K2**4 + 40*K2**2*K4 - 696*K2**2 - 176*K3**2 - 18*K4**2 + 728
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1501']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16516', 'vk6.16607', 'vk6.18086', 'vk6.18424', 'vk6.22947', 'vk6.23042', 'vk6.24533', 'vk6.24952', 'vk6.34920', 'vk6.35031', 'vk6.36676', 'vk6.37100', 'vk6.42493', 'vk6.42604', 'vk6.43952', 'vk6.44269', 'vk6.54743', 'vk6.54838', 'vk6.55902', 'vk6.56188', 'vk6.59207', 'vk6.59270', 'vk6.60428', 'vk6.60783', 'vk6.64759', 'vk6.64820', 'vk6.65544', 'vk6.65856', 'vk6.68059', 'vk6.68122', 'vk6.68622', 'vk6.68837', 'vk6.71380', 'vk6.71439', 'vk6.71902', 'vk6.71961', 'vk6.72819', 'vk6.74438', 'vk6.76627', 'vk6.77045', 'vk6.77758', 'vk6.78504', 'vk6.78637', 'vk6.78830', 'vk6.85160', 'vk6.87227', 'vk6.89277', 'vk6.89442']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3U1O4U5U2O6O5U3U4U6
R3 orbit {'O1O2O3U1O4U3U5O6U2O5U4U6', 'O1O2O3U1U4O5U2O4O6U3U6U5', 'O1O2O3U1O4U5U2O6O5U3U4U6'}
R3 orbit length 3
Gauss code of -K O1O2O3U4U5U1O6O4U2U6O5U3
Gauss code of K* O1O2O3U4U5U1O4U2O6O5U3U6
Gauss code of -K* O1O2O3U4U1O5O4U2O6U3U5U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 0 1 0 1],[ 2 0 1 2 2 1 1],[ 0 -1 0 0 1 0 1],[ 0 -2 0 0 1 0 1],[-1 -2 -1 -1 0 -1 0],[ 0 -1 0 0 1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -2],[-1 0 0 -1 -1 -1],[-1 0 0 -1 -1 -2],[ 0 1 1 0 0 -1],[ 0 1 1 0 0 -1],[ 2 1 2 1 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,0,0,2,0,1,1,1,1,1,2,0,1,1]
Phi over symmetry [-2,0,0,1,1,1,1,1,2,0,0,0,0,0,0]
Phi of -K [-2,0,0,1,1,1,1,1,2,0,0,0,0,0,0]
Phi of K* [-1,-1,0,0,2,0,0,0,1,0,0,2,0,1,1]
Phi of -K* [-2,0,0,1,1,1,1,1,2,0,1,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial 9w^2z+19w
Inner characteristic polynomial t^5+11t^3+2t
Outer characteristic polynomial t^6+17t^4+7t^2
Flat arrow polynomial -6*K1**2 + 3*K2 + 4
2-strand cable arrow polynomial 96*K1**4*K2 - 784*K1**4 - 512*K1**2*K2**2 + 1672*K1**2*K2 - 16*K1**2*K3**2 - 880*K1**2 + 568*K1*K2*K3 + 32*K1*K3*K4 - 56*K2**4 + 40*K2**2*K4 - 696*K2**2 - 176*K3**2 - 18*K4**2 + 728
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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