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Flat knot 6.437

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,0,1,3,2,-1,2,4,3,1,1,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.437']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 6*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.437']
Outer characteristic polynomial of the knot is: t^7+86t^5+335t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.437']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 640*K1**4*K2 - 2000*K1**4 + 160*K1**3*K2*K3 + 256*K1**2*K2**5 - 1920*K1**2*K2**4 + 1632*K1**2*K2**3 - 4560*K1**2*K2**2 + 4512*K1**2*K2 - 304*K1**2*K3**2 - 1936*K1**2 + 128*K1*K2**5*K3 + 1344*K1*K2**3*K3 + 3712*K1*K2*K3 + 464*K1*K3*K4 + 48*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1312*K2**6 - 320*K2**4*K3**2 - 192*K2**4*K4**2 + 1024*K2**4*K4 - 1720*K2**4 + 288*K2**3*K3*K5 + 64*K2**3*K4*K6 - 832*K2**2*K3**2 - 368*K2**2*K4**2 + 1208*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 834*K2**2 + 552*K2*K3*K5 + 96*K2*K4*K6 + 16*K2*K5*K7 + 16*K3**2*K6 - 1112*K3**2 - 456*K4**2 - 168*K5**2 - 30*K6**2 + 2398
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.437']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17014', 'vk6.17257', 'vk6.20232', 'vk6.21526', 'vk6.23428', 'vk6.23731', 'vk6.27443', 'vk6.29050', 'vk6.35502', 'vk6.35951', 'vk6.38863', 'vk6.41053', 'vk6.42926', 'vk6.43223', 'vk6.45618', 'vk6.47370', 'vk6.55201', 'vk6.55438', 'vk6.57068', 'vk6.58203', 'vk6.59592', 'vk6.59916', 'vk6.61597', 'vk6.62776', 'vk6.65004', 'vk6.65210', 'vk6.66696', 'vk6.67544', 'vk6.68282', 'vk6.68435', 'vk6.69348', 'vk6.70094']
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 O1O2O3O4O5U6U2O6U1U5U3U4
R3 orbit {'O1O2O3O4O5U6U2O6U1U5U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U1U5O6U4U6
Gauss code of K* O1O2O3O4U1U5U3U4U2O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U3U1U2U6U4
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 1 3 2 -1],[ 3 0 1 3 4 2 2],[ 2 -1 0 1 2 0 2],[-1 -3 -1 0 1 0 -1],[-3 -4 -2 -1 0 0 -3],[-2 -2 0 0 0 0 -2],[ 1 -2 -2 1 3 2 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 0 -1 -3 -2 -4],[-2 0 0 0 -2 0 -2],[-1 1 0 0 -1 -1 -3],[ 1 3 2 1 0 -2 -2],[ 2 2 0 1 2 0 -1],[ 3 4 2 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,0,1,3,2,4,0,2,0,2,1,1,3,2,2,1]
Phi over symmetry [-3,-2,-1,1,2,3,0,0,1,3,2,-1,2,4,3,1,1,1,1,1,1]
Phi of -K [-3,-2,-1,1,2,3,0,0,1,3,2,-1,2,4,3,1,1,1,1,1,1]
Phi of K* [-3,-2,-1,1,2,3,1,1,1,3,2,1,1,4,3,1,2,1,-1,0,0]
Phi of -K* [-3,-2,-1,1,2,3,1,2,3,2,4,2,1,0,2,1,2,3,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 10z+21
Enhanced Jones-Krushkal polynomial 4w^4z-8w^3z+14w^2z+21w
Inner characteristic polynomial t^6+58t^4+189t^2
Outer characteristic polynomial t^7+86t^5+335t^3
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 6*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 640*K1**4*K2 - 2000*K1**4 + 160*K1**3*K2*K3 + 256*K1**2*K2**5 - 1920*K1**2*K2**4 + 1632*K1**2*K2**3 - 4560*K1**2*K2**2 + 4512*K1**2*K2 - 304*K1**2*K3**2 - 1936*K1**2 + 128*K1*K2**5*K3 + 1344*K1*K2**3*K3 + 3712*K1*K2*K3 + 464*K1*K3*K4 + 48*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1312*K2**6 - 320*K2**4*K3**2 - 192*K2**4*K4**2 + 1024*K2**4*K4 - 1720*K2**4 + 288*K2**3*K3*K5 + 64*K2**3*K4*K6 - 832*K2**2*K3**2 - 368*K2**2*K4**2 + 1208*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 834*K2**2 + 552*K2*K3*K5 + 96*K2*K4*K6 + 16*K2*K5*K7 + 16*K3**2*K6 - 1112*K3**2 - 456*K4**2 - 168*K5**2 - 30*K6**2 + 2398
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}]]
If K is slice False
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