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Flat knot 6.415

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,2,1,3,4,1,1,2,3,1,1,2,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.415']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 4*K1**2 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.179', '6.278', '6.415']
Outer characteristic polynomial of the knot is: t^7+101t^5+74t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.415']
2-strand cable arrow polynomial of the knot is: -96*K1**4 - 768*K1**2*K2**6 + 1024*K1**2*K2**5 - 1088*K1**2*K2**4 + 224*K1**2*K2**3 - 352*K1**2*K2**2 + 568*K1**2*K2 - 32*K1**2*K3**2 - 464*K1**2 + 640*K1*K2**5*K3 + 224*K1*K2**3*K3 + 368*K1*K2*K3 + 64*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 544*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 64*K2**4*K4 + 400*K2**4 + 32*K2**3*K3*K5 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 72*K2**2*K4 - 256*K2**2 + 24*K2*K3*K5 - 136*K3**2 - 44*K4**2 - 8*K5**2 + 346
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.415']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16897', 'vk6.17140', 'vk6.19909', 'vk6.20206', 'vk6.21122', 'vk6.21489', 'vk6.23281', 'vk6.26808', 'vk6.26816', 'vk6.27388', 'vk6.28600', 'vk6.29015', 'vk6.35279', 'vk6.38242', 'vk6.38250', 'vk6.38811', 'vk6.40354', 'vk6.40989', 'vk6.42798', 'vk6.45105', 'vk6.45113', 'vk6.45562', 'vk6.46976', 'vk6.47344', 'vk6.55042', 'vk6.56679', 'vk6.56683', 'vk6.57752', 'vk6.58142', 'vk6.59426', 'vk6.61054', 'vk6.61070', 'vk6.62336', 'vk6.62721', 'vk6.64885', 'vk6.66378', 'vk6.67130', 'vk6.68198', 'vk6.69033', 'vk6.69826']
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 O1O2O3O4O5U6U1O6U2U3U4U5
R3 orbit {'O1O2O3O4O5U6U1O6U2U3U4U5', 'O1O2O3O4O5U2U6U1O6U3U4U5'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U1U2U3U4O6U5U6
Gauss code of K* O1O2O3O4U5U1U2U3U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U2U3U4U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -2 0 2 4 -1],[ 3 0 0 1 2 3 3],[ 2 0 0 1 2 3 2],[ 0 -1 -1 0 1 2 0],[-2 -2 -2 -1 0 1 -2],[-4 -3 -3 -2 -1 0 -4],[ 1 -3 -2 0 2 4 0]]
Primitive based matrix [[ 0 4 2 0 -1 -2 -3],[-4 0 -1 -2 -4 -3 -3],[-2 1 0 -1 -2 -2 -2],[ 0 2 1 0 0 -1 -1],[ 1 4 2 0 0 -2 -3],[ 2 3 2 1 2 0 0],[ 3 3 2 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,1,2,3,1,2,4,3,3,1,2,2,2,0,1,1,2,3,0]
Phi over symmetry [-4,-2,0,1,2,3,1,2,1,3,4,1,1,2,3,1,1,2,-1,-1,1]
Phi of -K [-3,-2,-1,0,2,4,1,-1,2,3,4,-1,1,2,3,1,1,1,1,2,1]
Phi of K* [-4,-2,0,1,2,3,1,2,1,3,4,1,1,2,3,1,1,2,-1,-1,1]
Phi of -K* [-3,-2,-1,0,2,4,0,3,1,2,3,2,1,2,3,0,2,4,1,2,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial 8w^4z-10w^3z+3w^2z+3w
Inner characteristic polynomial t^6+67t^4+20t^2
Outer characteristic polynomial t^7+101t^5+74t^3
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 4*K1**2 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial -96*K1**4 - 768*K1**2*K2**6 + 1024*K1**2*K2**5 - 1088*K1**2*K2**4 + 224*K1**2*K2**3 - 352*K1**2*K2**2 + 568*K1**2*K2 - 32*K1**2*K3**2 - 464*K1**2 + 640*K1*K2**5*K3 + 224*K1*K2**3*K3 + 368*K1*K2*K3 + 64*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 544*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 64*K2**4*K4 + 400*K2**4 + 32*K2**3*K3*K5 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 72*K2**2*K4 - 256*K2**2 + 24*K2*K3*K5 - 136*K3**2 - 44*K4**2 - 8*K5**2 + 346
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}]]
If K is slice False
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