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Flat knot 6.490

Min(phi) over symmetries of the knot is: [-4,0,0,0,2,2,0,2,3,3,4,1,1,0,0,0,0,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.490']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K2**2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.92', '6.348', '6.490', '6.494']
Outer characteristic polynomial of the knot is: t^7+76t^5+105t^3+25t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.490']
2-strand cable arrow polynomial of the knot is: -32*K2**4*K4**2 + 416*K2**4*K4 - 3488*K2**4 - 128*K2**3*K6 + 32*K2**2*K4**3 - 560*K2**2*K4**2 + 4088*K2**2*K4 - 644*K2**2 + 392*K2*K4*K6 - 8*K4**4 - 1192*K4**2 - 76*K6**2 + 1198
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.490']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72579', 'vk6.72582', 'vk6.72690', 'vk6.72697', 'vk6.73004', 'vk6.73010', 'vk6.73154', 'vk6.73158', 'vk6.73579', 'vk6.73581', 'vk6.74290', 'vk6.74295', 'vk6.74916', 'vk6.74920', 'vk6.75343', 'vk6.75346', 'vk6.76470', 'vk6.76477', 'vk6.77863', 'vk6.77903', 'vk6.77982', 'vk6.78006', 'vk6.79333', 'vk6.79338', 'vk6.80104', 'vk6.80106', 'vk6.80794', 'vk6.80796', 'vk6.85072', 'vk6.86708', 'vk6.87361', 'vk6.90156']
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 O1O2O3O4O5U1U4U3O6U5U2U6
R3 orbit {'O1O2O3O4O5U1U4U3O6U5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U1O6U3U2U5
Gauss code of K* O1O2O3U4O5O6O4U1U6U3U2U5
Gauss code of -K* O1O2O3U1O4O5O6U3U5U4U2U6
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 0 0 0 2 2],[ 4 0 4 2 1 3 2],[ 0 -4 0 -1 -1 2 2],[ 0 -2 1 0 0 2 1],[ 0 -1 1 0 0 1 1],[-2 -3 -2 -2 -1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 0 -4],[-2 0 1 -1 -2 -2 -3],[-2 -1 0 -1 -1 -2 -2],[ 0 1 1 0 0 1 -1],[ 0 2 1 0 0 1 -2],[ 0 2 2 -1 -1 0 -4],[ 4 3 2 1 2 4 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,0,4,-1,1,2,2,3,1,1,2,2,0,-1,1,-1,2,4]
Phi over symmetry [-4,0,0,0,2,2,0,2,3,3,4,1,1,0,0,0,0,1,1,1,-1]
Phi of -K [-4,0,0,0,2,2,0,2,3,3,4,1,1,0,0,0,0,1,1,1,-1]
Phi of K* [-2,-2,0,0,0,4,-1,0,1,1,4,0,0,1,3,-1,-1,0,0,2,3]
Phi of -K* [-4,0,0,0,2,2,1,2,4,2,3,0,1,1,1,1,1,2,2,2,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 2z^3+15z^2+30z+17
Enhanced Jones-Krushkal polynomial 2w^4z^3+15w^3z^2+30w^2z+17
Inner characteristic polynomial t^6+52t^4+25t^2+1
Outer characteristic polynomial t^7+76t^5+105t^3+25t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K2**2 + 3
2-strand cable arrow polynomial -32*K2**4*K4**2 + 416*K2**4*K4 - 3488*K2**4 - 128*K2**3*K6 + 32*K2**2*K4**3 - 560*K2**2*K4**2 + 4088*K2**2*K4 - 644*K2**2 + 392*K2*K4*K6 - 8*K4**4 - 1192*K4**2 - 76*K6**2 + 1198
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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