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Flat knot 6.823

Min(phi) over symmetries of the knot is: [-3,-2,-2,2,2,3,0,1,1,4,3,1,0,2,1,2,4,4,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.823']
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^7+105t^5+175t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.823']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 1920*K1**4 - 2624*K1**2*K2**2 + 8096*K1**2*K2 - 64*K1**2*K3**2 - 6880*K1**2 + 256*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 512*K1*K2*K3*K4 + 6464*K1*K2*K3 + 1408*K1*K3*K4 + 128*K1*K4*K5 - 416*K2**4 - 512*K2**2*K3**2 - 48*K2**2*K4**2 + 2032*K2**2*K4 - 6276*K2**2 + 576*K2*K3*K5 + 48*K2*K4*K6 - 2880*K3**2 - 1184*K4**2 - 128*K5**2 - 12*K6**2 + 5854
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.823']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72435', 'vk6.72497', 'vk6.72856', 'vk6.72909', 'vk6.74452', 'vk6.75065', 'vk6.76960', 'vk6.77792', 'vk6.77978', 'vk6.79460', 'vk6.79908', 'vk6.80929', 'vk6.87240', 'vk6.89370']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is a.
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4U2O5U1U6U5U4O6U3
R3 orbit {'O1O2O3O4U2O5U1U6U5U4O6U3'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* Same
Gauss code of -K* Same
Diagrammatic symmetry type a
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -2 2 3 2 -2],[ 3 0 0 4 3 1 1],[ 2 0 0 2 1 0 1],[-2 -4 -2 0 1 1 -4],[-3 -3 -1 -1 0 0 -4],[-2 -1 0 -1 0 0 -2],[ 2 -1 -1 4 4 2 0]]
Primitive based matrix [[ 0 3 2 2 -2 -2 -3],[-3 0 0 -1 -1 -4 -3],[-2 0 0 -1 0 -2 -1],[-2 1 1 0 -2 -4 -4],[ 2 1 0 2 0 1 0],[ 2 4 2 4 -1 0 -1],[ 3 3 1 4 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,2,2,3,0,1,1,4,3,1,0,2,1,2,4,4,-1,0,1]
Phi over symmetry [-3,-2,-2,2,2,3,0,1,1,4,3,1,0,2,1,2,4,4,-1,0,1]
Phi of -K [-3,-2,-2,2,2,3,0,1,1,4,3,1,0,2,1,2,4,4,-1,0,1]
Phi of K* [-3,-2,-2,2,2,3,0,1,1,4,3,1,0,2,1,2,4,4,-1,0,1]
Phi of -K* [-3,-2,-2,2,2,3,0,1,1,4,3,1,0,2,1,2,4,4,-1,0,1]
Symmetry type of based matrix a
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+22z+37
Enhanced Jones-Krushkal polynomial 2w^3z^2+22w^2z+37w
Inner characteristic polynomial t^6+71t^4+71t^2+1
Outer characteristic polynomial t^7+105t^5+175t^3+7t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial 128*K1**4*K2 - 1920*K1**4 - 2624*K1**2*K2**2 + 8096*K1**2*K2 - 64*K1**2*K3**2 - 6880*K1**2 + 256*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 512*K1*K2*K3*K4 + 6464*K1*K2*K3 + 1408*K1*K3*K4 + 128*K1*K4*K5 - 416*K2**4 - 512*K2**2*K3**2 - 48*K2**2*K4**2 + 2032*K2**2*K4 - 6276*K2**2 + 576*K2*K3*K5 + 48*K2*K4*K6 - 2880*K3**2 - 1184*K4**2 - 128*K5**2 - 12*K6**2 + 5854
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice True
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