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Flat knot 6.1794

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,2,-1,0,1,-1,0,0,1,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1794']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794']
Outer characteristic polynomial of the knot is: t^7+19t^5+52t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1794']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 832*K1**4*K2**2 + 1120*K1**4*K2 - 2960*K1**4 - 128*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 672*K1**3*K3 - 512*K1**2*K2**4 + 2464*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10912*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 12456*K1**2*K2 - 400*K1**2*K3**2 - 6904*K1**2 + 1024*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 416*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9864*K1*K2*K3 + 680*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 2664*K2**4 - 64*K2**3*K6 - 576*K2**2*K3**2 - 128*K2**2*K4**2 + 2472*K2**2*K4 - 4708*K2**2 + 464*K2*K3*K5 + 80*K2*K4*K6 - 2192*K3**2 - 498*K4**2 - 96*K5**2 - 12*K6**2 + 5392
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1794']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73272', 'vk6.73413', 'vk6.74012', 'vk6.74550', 'vk6.75180', 'vk6.75413', 'vk6.76028', 'vk6.76762', 'vk6.78141', 'vk6.78378', 'vk6.78989', 'vk6.79542', 'vk6.79974', 'vk6.80128', 'vk6.80509', 'vk6.80981', 'vk6.81876', 'vk6.82157', 'vk6.82182', 'vk6.82590', 'vk6.83583', 'vk6.83773', 'vk6.84048', 'vk6.84614', 'vk6.84941', 'vk6.85579', 'vk6.85720', 'vk6.85951', 'vk6.86732', 'vk6.87652', 'vk6.88943', 'vk6.89974']
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 O1O2O3U4U2O5O6U3U1O4U5U6
R3 orbit {'O1O2O3U4U2O5O6U3U1O4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U3U1O4O5U2U6
Gauss code of K* O1O2U3O4O5U2U6U1O3O6U4U5
Gauss code of -K* O1O2U3O4O5U1U2O6O3U5U6U4
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 -1 0 0 -1 0 2],[ 1 0 -1 1 -1 1 2],[ 0 1 0 1 -1 0 0],[ 0 -1 -1 0 0 0 1],[ 1 1 1 0 0 0 1],[ 0 -1 0 0 0 0 1],[-2 -2 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 0 0 0 -1 -1],[-2 0 0 -1 -1 -1 -2],[ 0 0 0 1 0 -1 1],[ 0 1 -1 0 0 0 -1],[ 0 1 0 0 0 0 -1],[ 1 1 1 0 0 0 1],[ 1 2 -1 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,0,0,0,1,1,0,1,1,1,2,-1,0,1,-1,0,0,1,0,1,-1]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,1,2,-1,0,1,-1,0,0,1,0,1,-1]
Phi of -K [-1,-1,0,0,0,2,-1,0,1,1,2,2,0,0,1,-1,0,2,0,1,1]
Phi of K* [-2,0,0,0,1,1,1,1,2,1,2,0,-1,0,1,0,0,1,2,0,-1]
Phi of -K* [-1,-1,0,0,0,2,-1,-1,1,1,2,1,0,0,1,0,1,0,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+13t^4+27t^2+4
Outer characteristic polynomial t^7+19t^5+52t^3+8t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial 128*K1**4*K2**3 - 832*K1**4*K2**2 + 1120*K1**4*K2 - 2960*K1**4 - 128*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 672*K1**3*K3 - 512*K1**2*K2**4 + 2464*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10912*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 12456*K1**2*K2 - 400*K1**2*K3**2 - 6904*K1**2 + 1024*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 416*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9864*K1*K2*K3 + 680*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 2664*K2**4 - 64*K2**3*K6 - 576*K2**2*K3**2 - 128*K2**2*K4**2 + 2472*K2**2*K4 - 4708*K2**2 + 464*K2*K3*K5 + 80*K2*K4*K6 - 2192*K3**2 - 498*K4**2 - 96*K5**2 - 12*K6**2 + 5392
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
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