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Flat knot 6.237

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,1,3,2,3,1,2,1,2,0,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.237']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.237', '6.602', '6.956', '6.986', '6.992', '6.1052', '6.1059']
Outer characteristic polynomial of the knot is: t^7+61t^5+34t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.237']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 160*K1**4*K2 - 464*K1**4 + 96*K1**3*K2*K3 + 512*K1**2*K2**3 - 2416*K1**2*K2**2 - 224*K1**2*K2*K4 + 3240*K1**2*K2 - 80*K1**2*K3**2 - 2472*K1**2 + 256*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2704*K1*K2*K3 + 408*K1*K3*K4 + 56*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 888*K2**4 - 304*K2**2*K3**2 - 200*K2**2*K4**2 + 1248*K2**2*K4 - 1834*K2**2 + 136*K2*K3*K5 + 80*K2*K4*K6 - 828*K3**2 - 458*K4**2 - 76*K5**2 - 30*K6**2 + 2048
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.237']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10131', 'vk6.10182', 'vk6.10325', 'vk6.10418', 'vk6.17675', 'vk6.17722', 'vk6.24238', 'vk6.24283', 'vk6.24814', 'vk6.25271', 'vk6.29910', 'vk6.29947', 'vk6.30010', 'vk6.30069', 'vk6.30962', 'vk6.31087', 'vk6.32140', 'vk6.32259', 'vk6.36603', 'vk6.36996', 'vk6.43607', 'vk6.43712', 'vk6.51687', 'vk6.51716', 'vk6.52040', 'vk6.52126', 'vk6.55709', 'vk6.55767', 'vk6.60279', 'vk6.60338', 'vk6.60659', 'vk6.61004', 'vk6.63328', 'vk6.63341', 'vk6.63370', 'vk6.63391', 'vk6.65453', 'vk6.65794', 'vk6.68551', 'vk6.68582']
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 O1O2O3O4O5U3O6U5U1U6U4U2
R3 orbit {'O1O2O3O4O5U3O6U5U1U6U4U2', 'O1O2O3O4O5U3U4O6U1U5U6U2'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U4U2U6U5U1O6U3
Gauss code of K* O1O2O3O4O5U2U5U6U4U1O6U3
Gauss code of -K* O1O2O3O4O5U3O6U5U2U6U1U4
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 1 -2 2 0 2],[ 3 0 3 -1 3 1 2],[-1 -3 0 -2 1 0 1],[ 2 1 2 0 2 1 1],[-2 -3 -1 -2 0 -1 1],[ 0 -1 0 -1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 -1 -1 -2],[-1 1 1 0 0 -2 -3],[ 0 1 1 0 0 -1 -1],[ 2 2 1 2 1 0 1],[ 3 3 2 3 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,-1,1,1,2,3,1,1,1,2,0,2,3,1,1,-1]
Phi over symmetry [-3,-2,0,1,2,2,-1,1,3,2,3,1,2,1,2,0,1,1,1,1,-1]
Phi of -K [-3,-2,0,1,2,2,2,2,1,2,3,1,1,2,3,1,1,1,0,0,-1]
Phi of K* [-2,-2,-1,0,2,3,-1,0,1,3,3,0,1,2,2,1,1,1,1,2,2]
Phi of -K* [-3,-2,0,1,2,2,-1,1,3,2,3,1,2,1,2,0,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial 3w^3z^2+16w^2z+21w
Inner characteristic polynomial t^6+39t^4+17t^2
Outer characteristic polynomial t^7+61t^5+34t^3+3t
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -64*K1**4*K2**2 + 160*K1**4*K2 - 464*K1**4 + 96*K1**3*K2*K3 + 512*K1**2*K2**3 - 2416*K1**2*K2**2 - 224*K1**2*K2*K4 + 3240*K1**2*K2 - 80*K1**2*K3**2 - 2472*K1**2 + 256*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2704*K1*K2*K3 + 408*K1*K3*K4 + 56*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 888*K2**4 - 304*K2**2*K3**2 - 200*K2**2*K4**2 + 1248*K2**2*K4 - 1834*K2**2 + 136*K2*K3*K5 + 80*K2*K4*K6 - 828*K3**2 - 458*K4**2 - 76*K5**2 - 30*K6**2 + 2048
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {4}, {1, 3}, {2}]]
If K is slice False
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