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Flat knot 6.497

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,0,2,2,4,4,1,0,1,0,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.497']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.223', '6.281', '6.497']
Outer characteristic polynomial of the knot is: t^7+65t^5+74t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.497']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 832*K1**4*K2 - 1408*K1**4 + 544*K1**2*K2**3 - 2688*K1**2*K2**2 + 3632*K1**2*K2 - 336*K1**2*K3**2 - 128*K1**2*K4**2 - 2916*K1**2 + 160*K1*K2**3*K3 + 2712*K1*K2*K3 + 1080*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 760*K2**4 - 432*K2**2*K3**2 - 288*K2**2*K4**2 + 840*K2**2*K4 - 2078*K2**2 + 424*K2*K3*K5 + 152*K2*K4*K6 - 80*K3**4 - 96*K3**2*K4**2 + 48*K3**2*K6 - 1208*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 814*K4**2 - 244*K5**2 - 42*K6**2 - 8*K7**2 - 2*K8**2 + 2862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.497']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17103', 'vk6.17345', 'vk6.20590', 'vk6.21997', 'vk6.23490', 'vk6.23829', 'vk6.28052', 'vk6.29509', 'vk6.35635', 'vk6.36077', 'vk6.39470', 'vk6.41669', 'vk6.43000', 'vk6.43311', 'vk6.46054', 'vk6.47720', 'vk6.55242', 'vk6.55493', 'vk6.57460', 'vk6.58625', 'vk6.59640', 'vk6.59988', 'vk6.62131', 'vk6.63095', 'vk6.65036', 'vk6.65236', 'vk6.66992', 'vk6.67855', 'vk6.68302', 'vk6.68451', 'vk6.69607', 'vk6.70298']
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 O1O2O3O4O5U1U5U3O6U4U2U6
R3 orbit {'O1O2O3O4O5U1U5U3O6U4U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U2O6U3U1U5
Gauss code of K* O1O2O3U4O5O6O4U1U6U3U5U2
Gauss code of -K* O1O2O3U1O4O5O6U5U3U4U2U6
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 -4 0 0 1 1 2],[ 4 0 4 2 3 1 2],[ 0 -4 0 -1 1 0 2],[ 0 -2 1 0 1 0 1],[-1 -3 -1 -1 0 0 1],[-1 -1 0 0 0 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 0 -4],[-2 0 0 -1 -1 -2 -2],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -1 -3],[ 0 1 0 1 0 1 -2],[ 0 2 0 1 -1 0 -4],[ 4 2 1 3 2 4 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,0,4,0,1,1,2,2,0,0,0,1,1,1,3,-1,2,4]
Phi over symmetry [-4,0,0,1,1,2,0,2,2,4,4,1,0,1,0,0,1,1,0,0,1]
Phi of -K [-4,0,0,1,1,2,0,2,2,4,4,1,0,1,0,0,1,1,0,0,1]
Phi of K* [-2,-1,-1,0,0,4,0,1,0,1,4,0,0,0,2,1,1,4,-1,0,2]
Phi of -K* [-4,0,0,1,1,2,2,4,1,3,2,1,0,1,1,0,1,2,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial -2w^3z+15w^2z+27w
Inner characteristic polynomial t^6+43t^4+15t^2
Outer characteristic polynomial t^7+65t^5+74t^3
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
2-strand cable arrow polynomial -256*K1**4*K2**2 + 832*K1**4*K2 - 1408*K1**4 + 544*K1**2*K2**3 - 2688*K1**2*K2**2 + 3632*K1**2*K2 - 336*K1**2*K3**2 - 128*K1**2*K4**2 - 2916*K1**2 + 160*K1*K2**3*K3 + 2712*K1*K2*K3 + 1080*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 760*K2**4 - 432*K2**2*K3**2 - 288*K2**2*K4**2 + 840*K2**2*K4 - 2078*K2**2 + 424*K2*K3*K5 + 152*K2*K4*K6 - 80*K3**4 - 96*K3**2*K4**2 + 48*K3**2*K6 - 1208*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 814*K4**2 - 244*K5**2 - 42*K6**2 - 8*K7**2 - 2*K8**2 + 2862
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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