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Flat knot 6.69

Min(phi) over symmetries of the knot is: [-4,-3,-2,2,3,4,0,1,4,3,5,0,2,1,3,3,2,4,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.69']
Arrow polynomial of the knot is: -16*K1**4 + 8*K1**3 + 8*K1**2*K2 + 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.69', '6.70']
Outer characteristic polynomial of the knot is: t^7+157t^5+337t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.69']
2-strand cable arrow polynomial of the knot is: -1312*K1**4 - 64*K1**3*K3 + 256*K1**2*K2**5 - 2304*K1**2*K2**4 + 3584*K1**2*K2**3 - 10112*K1**2*K2**2 - 384*K1**2*K2*K4 + 9568*K1**2*K2 - 64*K1**2*K3**2 - 5464*K1**2 + 256*K1*K2**5*K3 + 4288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 3008*K1*K2**2*K3 - 576*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 8368*K1*K2*K3 + 512*K1*K3*K4 + 48*K1*K4*K5 - 256*K2**8 + 256*K2**6*K4 - 3136*K2**6 - 768*K2**4*K3**2 - 192*K2**4*K4**2 + 3072*K2**4*K4 - 5232*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 - 576*K2**3*K6 - 2080*K2**2*K3**2 - 496*K2**2*K4**2 + 4160*K2**2*K4 - 64*K2**2*K5**2 - 1456*K2**2 + 624*K2*K3*K5 + 112*K2*K4*K6 - 1760*K3**2 - 500*K4**2 - 56*K5**2 - 8*K6**2 + 4186
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.69']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81914', 'vk6.81963', 'vk6.82116', 'vk6.82130', 'vk6.82640', 'vk6.82692', 'vk6.82795', 'vk6.82798', 'vk6.83071', 'vk6.83082', 'vk6.83538', 'vk6.84668', 'vk6.84992', 'vk6.85845', 'vk6.86212', 'vk6.88476', 'vk6.89081', 'vk6.89091', 'vk6.89648', 'vk6.90060']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4O5O6U2U3U1U5U6U4
R3 orbit {'O1O2O3O4O5O6U2U3U1U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U1U2U6U4U5
Gauss code of K* O1O2O3O4O5O6U3U1U2U6U4U5
Gauss code of -K* Same
Diagrammatic symmetry type -
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 -4 -2 3 2 4],[ 3 0 -1 1 5 3 4],[ 4 1 0 1 4 2 3],[ 2 -1 -1 0 3 1 2],[-3 -5 -4 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-4 -4 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 4 3 2 -2 -3 -4],[-4 0 -1 -1 -2 -4 -3],[-3 1 0 -1 -3 -5 -4],[-2 1 1 0 -1 -3 -2],[ 2 2 3 1 0 -1 -1],[ 3 4 5 3 1 0 -1],[ 4 3 4 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-2,2,3,4,1,1,2,4,3,1,3,5,4,1,3,2,1,1,1]
Phi over symmetry [-4,-3,-2,2,3,4,0,1,4,3,5,0,2,1,3,3,2,4,0,1,0]
Phi of -K [-4,-3,-2,2,3,4,0,1,4,3,5,0,2,1,3,3,2,4,0,1,0]
Phi of K* [-4,-3,-2,2,3,4,0,1,4,3,5,0,2,1,3,3,2,4,0,1,0]
Phi of -K* [-4,-3,-2,2,3,4,1,1,2,4,3,1,3,5,4,1,3,2,1,1,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial -4w^4z^2+10w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+99t^4+47t^2+1
Outer characteristic polynomial t^7+157t^5+337t^3+13t
Flat arrow polynomial -16*K1**4 + 8*K1**3 + 8*K1**2*K2 + 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial -1312*K1**4 - 64*K1**3*K3 + 256*K1**2*K2**5 - 2304*K1**2*K2**4 + 3584*K1**2*K2**3 - 10112*K1**2*K2**2 - 384*K1**2*K2*K4 + 9568*K1**2*K2 - 64*K1**2*K3**2 - 5464*K1**2 + 256*K1*K2**5*K3 + 4288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 3008*K1*K2**2*K3 - 576*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 8368*K1*K2*K3 + 512*K1*K3*K4 + 48*K1*K4*K5 - 256*K2**8 + 256*K2**6*K4 - 3136*K2**6 - 768*K2**4*K3**2 - 192*K2**4*K4**2 + 3072*K2**4*K4 - 5232*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 - 576*K2**3*K6 - 2080*K2**2*K3**2 - 496*K2**2*K4**2 + 4160*K2**2*K4 - 64*K2**2*K5**2 - 1456*K2**2 + 624*K2*K3*K5 + 112*K2*K4*K6 - 1760*K3**2 - 500*K4**2 - 56*K5**2 - 8*K6**2 + 4186
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice True
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