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Flat knot 6.650

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,3,2,3,1,2,1,2,1,2,2,2,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.650']
Arrow polynomial of the knot is: -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.574', '6.593', '6.604', '6.649', '6.650', '6.673', '6.690', '6.783', '6.973', '6.985', '6.1033', '6.1035']
Outer characteristic polynomial of the knot is: t^7+73t^5+38t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.650']
2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 2240*K1**4 + 128*K1**3*K2*K3 - 864*K1**3*K3 - 1856*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 7000*K1**2*K2 - 736*K1**2*K3**2 - 64*K1**2*K4**2 - 5464*K1**2 - 224*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 5192*K1*K2*K3 + 1168*K1*K3*K4 + 80*K1*K4*K5 - 88*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 416*K2**2*K4 - 4022*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 2016*K3**2 - 498*K4**2 - 40*K5**2 - 2*K6**2 + 4192
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.650']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73283', 'vk6.73285', 'vk6.73426', 'vk6.73428', 'vk6.74098', 'vk6.74102', 'vk6.74669', 'vk6.74673', 'vk6.75428', 'vk6.75430', 'vk6.76136', 'vk6.76140', 'vk6.78164', 'vk6.78166', 'vk6.78396', 'vk6.78398', 'vk6.79104', 'vk6.79108', 'vk6.79987', 'vk6.79989', 'vk6.80140', 'vk6.80142', 'vk6.80612', 'vk6.80616', 'vk6.83807', 'vk6.83823', 'vk6.85110', 'vk6.85133', 'vk6.86605', 'vk6.86610', 'vk6.87381', 'vk6.87383']
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 O1O2O3O4U2O5U1U3O6U4U5U6
R3 orbit {'O1O2O3O4U2O5U1U3O6U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1O5U2U4O6U3
Gauss code of K* O1O2O3U4U5U6U1O5U2O4O6U3
Gauss code of -K* O1O2O3U1O4O5U2O6U3U4U6U5
Diagrammatic symmetry type c
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 0 1 2 2],[ 3 0 0 2 3 3 2],[ 2 0 0 1 2 2 1],[ 0 -2 -1 0 1 2 2],[-1 -3 -2 -1 0 1 2],[-2 -3 -2 -2 -1 0 1],[-2 -2 -1 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 1 -1 -2 -2 -3],[-2 -1 0 -2 -2 -1 -2],[-1 1 2 0 -1 -2 -3],[ 0 2 2 1 0 -1 -2],[ 2 2 1 2 1 0 0],[ 3 3 2 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,-1,1,2,2,3,2,2,1,2,1,2,3,1,2,0]
Phi over symmetry [-3,-2,0,1,2,2,0,2,3,2,3,1,2,1,2,1,2,2,2,1,-1]
Phi of -K [-3,-2,0,1,2,2,1,1,1,2,3,1,1,2,3,0,0,0,0,-1,-1]
Phi of K* [-2,-2,-1,0,2,3,-1,-1,0,3,3,0,0,2,2,0,1,1,1,1,1]
Phi of -K* [-3,-2,0,1,2,2,0,2,3,2,3,1,2,1,2,1,2,2,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+51t^4+11t^2
Outer characteristic polynomial t^7+73t^5+38t^3+4t
Flat arrow polynomial -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial 160*K1**4*K2 - 2240*K1**4 + 128*K1**3*K2*K3 - 864*K1**3*K3 - 1856*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 7000*K1**2*K2 - 736*K1**2*K3**2 - 64*K1**2*K4**2 - 5464*K1**2 - 224*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 5192*K1*K2*K3 + 1168*K1*K3*K4 + 80*K1*K4*K5 - 88*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 416*K2**2*K4 - 4022*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 2016*K3**2 - 498*K4**2 - 40*K5**2 - 2*K6**2 + 4192
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}]]
If K is slice False
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