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Flat knot 6.810

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,1,1,2,4,0,0,0,1,0,1,2,0,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.810']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.206', '6.236', '6.575', '6.580', '6.613', '6.619', '6.810', '6.819', '6.831', '6.838', '6.957', '6.1018', '6.1028', '6.1046', '6.1073', '6.1279', '6.1507', '6.1532', '6.1556', '6.1639', '6.1688', '6.1924', '6.1931']
Outer characteristic polynomial of the knot is: t^7+85t^5+117t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.810']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 512*K1**4*K2**3 - 1792*K1**4*K2**2 + 2240*K1**4*K2 - 2592*K1**4 + 960*K1**3*K2*K3 - 128*K1**3*K3 - 896*K1**2*K2**4 + 2816*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 8896*K1**2*K2**2 - 448*K1**2*K2*K4 + 7712*K1**2*K2 - 224*K1**2*K3**2 - 3576*K1**2 + 1152*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 + 5472*K1*K2*K3 + 112*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 1936*K2**4 - 544*K2**2*K3**2 - 48*K2**2*K4**2 + 832*K2**2*K4 - 1728*K2**2 + 64*K2*K3*K5 - 920*K3**2 - 44*K4**2 + 2922
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.810']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10921', 'vk6.10937', 'vk6.12086', 'vk6.12103', 'vk6.14474', 'vk6.15695', 'vk6.16139', 'vk6.30530', 'vk6.30562', 'vk6.31807', 'vk6.34065', 'vk6.34154', 'vk6.34501', 'vk6.51766', 'vk6.52635', 'vk6.54142', 'vk6.54335', 'vk6.54538', 'vk6.63471', 'vk6.63484']
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 O1O2O3O4U1O5U3U4U6U5O6U2
R3 orbit {'O1O2O3O4U1O5U3U4U6U5O6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3O5U6U5U1U2O6U4
Gauss code of K* O1O2O3O4U3O5U6U5U1U2O6U4
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 1 -1 1 3 -1],[ 3 0 3 1 2 2 3],[-1 -3 0 -2 0 3 -2],[ 1 -1 2 0 1 2 0],[-1 -2 0 -1 0 1 -2],[-3 -2 -3 -2 -1 0 -3],[ 1 -3 2 0 2 3 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 -1 -3 -2 -3 -2],[-1 1 0 0 -1 -2 -2],[-1 3 0 0 -2 -2 -3],[ 1 2 1 2 0 0 -1],[ 1 3 2 2 0 0 -3],[ 3 2 2 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,1,3,2,3,2,0,1,2,2,2,2,3,0,1,3]
Phi over symmetry [-3,-1,-1,1,1,3,-1,1,1,2,4,0,0,0,1,0,1,2,0,-1,1]
Phi of -K [-3,-1,-1,1,1,3,-1,1,1,2,4,0,0,0,1,0,1,2,0,-1,1]
Phi of K* [-3,-1,-1,1,1,3,-1,1,1,2,4,0,0,0,1,0,1,2,0,-1,1]
Phi of -K* [-3,-1,-1,1,1,3,1,3,2,3,2,0,1,2,2,2,2,3,0,1,3]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+63t^4+85t^2+4
Outer characteristic polynomial t^7+85t^5+117t^3+6t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial -128*K1**6 + 512*K1**4*K2**3 - 1792*K1**4*K2**2 + 2240*K1**4*K2 - 2592*K1**4 + 960*K1**3*K2*K3 - 128*K1**3*K3 - 896*K1**2*K2**4 + 2816*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 8896*K1**2*K2**2 - 448*K1**2*K2*K4 + 7712*K1**2*K2 - 224*K1**2*K3**2 - 3576*K1**2 + 1152*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 + 5472*K1*K2*K3 + 112*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 1936*K2**4 - 544*K2**2*K3**2 - 48*K2**2*K4**2 + 832*K2**2*K4 - 1728*K2**2 + 64*K2*K3*K5 - 920*K3**2 - 44*K4**2 + 2922
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice True
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