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Flat knot 6.726

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,1,3,3,0,0,1,1,0,1,2,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.726']
Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384']
Outer characteristic polynomial of the knot is: t^7+68t^5+66t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.726']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 128*K1**4*K2**2 + 288*K1**4*K2 - 1280*K1**4 + 64*K1**3*K2*K3 - 1808*K1**2*K2**2 + 2320*K1**2*K2 - 192*K1**2*K3**2 - 584*K1**2 + 1688*K1*K2*K3 + 232*K1*K3*K4 + 8*K1*K4*K5 - 592*K2**4 - 176*K2**2*K3**2 - 8*K2**2*K4**2 + 456*K2**2*K4 - 646*K2**2 + 152*K2*K3*K5 + 8*K2*K4*K6 - 420*K3**2 - 152*K4**2 - 36*K5**2 - 2*K6**2 + 934
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.726']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11056', 'vk6.11136', 'vk6.12218', 'vk6.12327', 'vk6.16429', 'vk6.19218', 'vk6.19327', 'vk6.19511', 'vk6.19620', 'vk6.22732', 'vk6.22833', 'vk6.26028', 'vk6.26091', 'vk6.26412', 'vk6.26513', 'vk6.30625', 'vk6.30722', 'vk6.31929', 'vk6.34782', 'vk6.35500', 'vk6.35949', 'vk6.38098', 'vk6.38131', 'vk6.38864', 'vk6.41054', 'vk6.42397', 'vk6.42924', 'vk6.43221', 'vk6.44617', 'vk6.44753', 'vk6.45617', 'vk6.51848', 'vk6.52811', 'vk6.55203', 'vk6.58204', 'vk6.59594', 'vk6.62775', 'vk6.64727', 'vk6.66277', 'vk6.66295']
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 O1O2O3O4U1O5U3U6U4O6U5U2
R3 orbit {'O1O2O3O4U1O5U3U6U4O6U5U2', 'O1O2O3O4U1U2O5U6U4O6U3U5'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U3U5O6U1U6U2O5U4
Gauss code of K* O1O2O3U2O4O5U6U5U1U3O6U4
Gauss code of -K* O1O2O3U4O5U1U3U6U5O6O4U2
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 1 -1 2 2 -1],[ 3 0 3 1 2 2 2],[-1 -3 0 -2 1 2 -2],[ 1 -1 2 0 1 2 0],[-2 -2 -1 -1 0 0 -2],[-2 -2 -2 -2 0 0 -2],[ 1 -2 2 0 2 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 -1 -2 -2],[-2 0 0 -2 -2 -2 -2],[-1 1 2 0 -2 -2 -3],[ 1 1 2 2 0 0 -1],[ 1 2 2 2 0 0 -2],[ 3 2 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,1,1,2,2,2,2,2,2,2,2,3,0,1,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,1,3,3,0,0,1,1,0,1,2,-1,0,0]
Phi of -K [-3,-1,-1,1,2,2,0,1,1,3,3,0,0,1,1,0,1,2,-1,0,0]
Phi of K* [-2,-2,-1,1,1,3,0,-1,1,1,3,0,1,2,3,0,0,1,0,0,1]
Phi of -K* [-3,-1,-1,1,2,2,1,2,3,2,2,0,2,1,2,2,2,2,1,2,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 10z+21
Enhanced Jones-Krushkal polynomial 10w^2z+21w
Inner characteristic polynomial t^6+48t^4+36t^2
Outer characteristic polynomial t^7+68t^5+66t^3
Flat arrow polynomial -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -128*K1**6 - 128*K1**4*K2**2 + 288*K1**4*K2 - 1280*K1**4 + 64*K1**3*K2*K3 - 1808*K1**2*K2**2 + 2320*K1**2*K2 - 192*K1**2*K3**2 - 584*K1**2 + 1688*K1*K2*K3 + 232*K1*K3*K4 + 8*K1*K4*K5 - 592*K2**4 - 176*K2**2*K3**2 - 8*K2**2*K4**2 + 456*K2**2*K4 - 646*K2**2 + 152*K2*K3*K5 + 8*K2*K4*K6 - 420*K3**2 - 152*K4**2 - 36*K5**2 - 2*K6**2 + 934
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}]]
If K is slice False
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