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Flat knot 6.531

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,0,3,2,4,0,1,1,2,1,0,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.531']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.330', '6.531', '6.1076', '6.1079', '6.1567']
Outer characteristic polynomial of the knot is: t^7+90t^5+90t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.531']
2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 448*K1**2*K2**4 + 1216*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7440*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7456*K1**2*K2 - 80*K1**2*K3**2 - 112*K1**2*K4**2 - 5896*K1**2 + 1440*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 416*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7920*K1*K2*K3 + 952*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 224*K2**6 + 832*K2**4*K4 - 3952*K2**4 - 224*K2**3*K6 - 944*K2**2*K3**2 - 672*K2**2*K4**2 + 4144*K2**2*K4 - 3782*K2**2 + 584*K2*K3*K5 + 376*K2*K4*K6 - 2148*K3**2 - 1244*K4**2 - 108*K5**2 - 58*K6**2 + 4802
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.531']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11076', 'vk6.11156', 'vk6.12242', 'vk6.12351', 'vk6.18322', 'vk6.18659', 'vk6.24756', 'vk6.25213', 'vk6.30659', 'vk6.30752', 'vk6.31887', 'vk6.31957', 'vk6.36942', 'vk6.37407', 'vk6.44137', 'vk6.44459', 'vk6.51865', 'vk6.51912', 'vk6.52732', 'vk6.52841', 'vk6.56110', 'vk6.56331', 'vk6.60627', 'vk6.60962', 'vk6.63525', 'vk6.63571', 'vk6.64007', 'vk6.64053', 'vk6.65761', 'vk6.66024', 'vk6.68769', 'vk6.68978']
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 O1O2O3O4O5U2U6U4O6U1U5U3
R3 orbit {'O1O2O3O4O5U2U6U4O6U1U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U5O6U2U6U4
Gauss code of K* O1O2O3U2O4O5O6U4U1U6U3U5
Gauss code of -K* O1O2O3U4O5O4O6U2U5U1U6U3
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 -2 -3 2 1 3 -1],[ 2 0 -1 3 2 3 1],[ 3 1 0 3 1 2 2],[-2 -3 -3 0 0 1 -3],[-1 -2 -1 0 0 0 -1],[-3 -3 -2 -1 0 0 -3],[ 1 -1 -2 3 1 3 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 0 -3 -3 -2],[-2 1 0 0 -3 -3 -3],[-1 0 0 0 -1 -2 -1],[ 1 3 3 1 0 -1 -2],[ 2 3 3 2 1 0 -1],[ 3 2 3 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,1,0,3,3,2,0,3,3,3,1,2,1,1,2,1]
Phi over symmetry [-3,-2,-1,1,2,3,0,0,3,2,4,0,1,1,2,1,0,1,1,2,0]
Phi of -K [-3,-2,-1,1,2,3,0,0,3,2,4,0,1,1,2,1,0,1,1,2,0]
Phi of K* [-3,-2,-1,1,2,3,0,2,1,2,4,1,0,1,2,1,1,3,0,0,0]
Phi of -K* [-3,-2,-1,1,2,3,1,2,1,3,2,1,2,3,3,1,3,3,0,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2-4w^3z+27w^2z+23w
Inner characteristic polynomial t^6+62t^4+46t^2+1
Outer characteristic polynomial t^7+90t^5+90t^3+13t
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 448*K1**2*K2**4 + 1216*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7440*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7456*K1**2*K2 - 80*K1**2*K3**2 - 112*K1**2*K4**2 - 5896*K1**2 + 1440*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 416*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7920*K1*K2*K3 + 952*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 224*K2**6 + 832*K2**4*K4 - 3952*K2**4 - 224*K2**3*K6 - 944*K2**2*K3**2 - 672*K2**2*K4**2 + 4144*K2**2*K4 - 3782*K2**2 + 584*K2*K3*K5 + 376*K2*K4*K6 - 2148*K3**2 - 1244*K4**2 - 108*K5**2 - 58*K6**2 + 4802
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice True
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