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Flat knot 6.411

Min(phi) over symmetries of the knot is: [-2,-1,1,2,-1,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.411', '7.20042', '7.29751']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717']
Outer characteristic polynomial of the knot is: t^5+19t^3+20t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.411', '7.20042', '7.29751']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 448*K1**4*K2**2 + 640*K1**4*K2 - 1456*K1**4 + 256*K1**3*K2*K3 - 2080*K1**2*K2**2 + 2288*K1**2*K2 - 912*K1**2*K3**2 - 224*K1**2*K4**2 + 16*K1**2 + 2240*K1*K2*K3 + 704*K1*K3*K4 + 128*K1*K4*K5 - 624*K2**4 - 640*K2**2*K3**2 - 176*K2**2*K4**2 + 496*K2**2*K4 - 316*K2**2 + 368*K2*K3*K5 + 96*K2*K4*K6 - 336*K3**2 - 132*K4**2 - 32*K5**2 - 4*K6**2 + 658
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.411']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.63', 'vk6.118', 'vk6.213', 'vk6.260', 'vk6.303', 'vk6.685', 'vk6.712', 'vk6.759', 'vk6.1215', 'vk6.1262', 'vk6.1351', 'vk6.1398', 'vk6.1504', 'vk6.1579', 'vk6.1929', 'vk6.2052', 'vk6.2447', 'vk6.2476', 'vk6.2646', 'vk6.2989', 'vk6.5740', 'vk6.5771', 'vk6.7805', 'vk6.7836', 'vk6.10265', 'vk6.10410', 'vk6.13299', 'vk6.13332', 'vk6.14778', 'vk6.14818', 'vk6.15934', 'vk6.15974', 'vk6.18059', 'vk6.24497', 'vk6.25846', 'vk6.33048', 'vk6.37399', 'vk6.37957', 'vk6.38020', 'vk6.44864']
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 O1O2O3O4O5U4U5O6U2U6U1U3
R3 orbit {'O1O2O3O4O5U4U5U1O6U2U6U3', 'O1O2O3O4O5U4U5O6U2U6U1U3'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U3U5U6U4O6U1U2
Gauss code of K* O1O2O3O4U3U1U4U5U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U6U1U4U2
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 -1 -2 2 -1 1 1],[ 1 0 -1 2 -1 1 1],[ 2 1 0 2 -1 1 1],[-2 -2 -2 0 -1 1 0],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 1 -1 -2],[-2 0 1 -1 -2],[-1 -1 0 -1 -1],[ 1 1 1 0 1],[ 2 2 1 -1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,1,2,-1,1,2,1,1,-1]
Phi over symmetry [-2,-1,1,2,-1,1,2,1,1,-1]
Phi of -K [-2,-1,1,2,2,2,2,1,2,2]
Phi of K* [-2,-1,1,2,2,2,2,1,2,2]
Phi of -K* [-2,-1,1,2,-1,1,2,1,1,-1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 10z+21
Enhanced Jones-Krushkal polynomial 10w^2z+21w
Inner characteristic polynomial t^4+9t^2+4
Outer characteristic polynomial t^5+19t^3+20t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -512*K1**6 - 448*K1**4*K2**2 + 640*K1**4*K2 - 1456*K1**4 + 256*K1**3*K2*K3 - 2080*K1**2*K2**2 + 2288*K1**2*K2 - 912*K1**2*K3**2 - 224*K1**2*K4**2 + 16*K1**2 + 2240*K1*K2*K3 + 704*K1*K3*K4 + 128*K1*K4*K5 - 624*K2**4 - 640*K2**2*K3**2 - 176*K2**2*K4**2 + 496*K2**2*K4 - 316*K2**2 + 368*K2*K3*K5 + 96*K2*K4*K6 - 336*K3**2 - 132*K4**2 - 32*K5**2 - 4*K6**2 + 658
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}]]
If K is slice True
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