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Flat knot 6.465

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,2,3,0,1,0,1,0,1,1,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.465']
Arrow polynomial of the knot is: 12*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 6*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.465']
Outer characteristic polynomial of the knot is: t^7+57t^5+43t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.465']
2-strand cable arrow polynomial of the knot is: -768*K1**4*K2**2 + 1984*K1**4*K2 - 3616*K1**4 - 512*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 992*K1**3*K3 - 384*K1**2*K2**4 - 512*K1**2*K2**3*K4 + 3232*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 + 896*K1**2*K2**2*K4 - 12784*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1664*K1**2*K2*K4 + 12752*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 7160*K1**2 - 256*K1*K2**3*K3*K4 + 3520*K1*K2**3*K3 + 832*K1*K2**2*K3*K4 - 2496*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 928*K1*K2**2*K5 + 32*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 11376*K1*K2*K3 + 1360*K1*K3*K4 + 128*K1*K4*K5 + 16*K1*K5*K6 - 96*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1056*K2**4*K4 - 4064*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 1952*K2**2*K3**2 - 648*K2**2*K4**2 + 2968*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 4176*K2**2 + 840*K2*K3*K5 + 120*K2*K4*K6 + 8*K3**2*K6 - 2644*K3**2 - 736*K4**2 - 116*K5**2 - 24*K6**2 + 5806
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.465']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4751', 'vk6.5080', 'vk6.6297', 'vk6.6738', 'vk6.8262', 'vk6.8713', 'vk6.9644', 'vk6.9961', 'vk6.20392', 'vk6.21735', 'vk6.27724', 'vk6.29268', 'vk6.39170', 'vk6.41396', 'vk6.45896', 'vk6.47539', 'vk6.48791', 'vk6.49004', 'vk6.49615', 'vk6.49820', 'vk6.50819', 'vk6.51036', 'vk6.51294', 'vk6.51491', 'vk6.57261', 'vk6.58480', 'vk6.61907', 'vk6.63014', 'vk6.66878', 'vk6.67754', 'vk6.69508', 'vk6.70224']
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 O1O2O3O4O5U6U4O6U1U5U2U3
R3 orbit {'O1O2O3O4O5U6U4O6U1U5U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U4U1U5O6U2U6
Gauss code of K* O1O2O3O4U1U3U4U5U2O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U3U6U1U2U4
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 -3 0 2 0 2 -1],[ 3 0 2 3 1 2 2],[ 0 -2 0 1 0 1 -1],[-2 -3 -1 0 0 1 -3],[ 0 -1 0 0 0 0 0],[-2 -2 -1 -1 0 0 -2],[ 1 -2 1 3 0 2 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 1 0 -1 -3 -3],[-2 -1 0 0 -1 -2 -2],[ 0 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -2],[ 1 3 2 0 1 0 -2],[ 3 3 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-1,0,1,3,3,0,1,2,2,0,0,1,1,2,2]
Phi over symmetry [-3,-1,0,0,2,2,0,1,2,2,3,0,1,0,1,0,1,1,2,2,-1]
Phi of -K [-3,-1,0,0,2,2,0,1,2,2,3,0,1,0,1,0,1,1,2,2,-1]
Phi of K* [-2,-2,0,0,1,3,-1,1,2,1,3,1,2,0,2,0,0,1,1,2,0]
Phi of -K* [-3,-1,0,0,2,2,2,1,2,2,3,0,1,2,3,0,0,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+39t^4+23t^2
Outer characteristic polynomial t^7+57t^5+43t^3+4t
Flat arrow polynomial 12*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 6*K1 + 4*K2 + 5
2-strand cable arrow polynomial -768*K1**4*K2**2 + 1984*K1**4*K2 - 3616*K1**4 - 512*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 992*K1**3*K3 - 384*K1**2*K2**4 - 512*K1**2*K2**3*K4 + 3232*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 + 896*K1**2*K2**2*K4 - 12784*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1664*K1**2*K2*K4 + 12752*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 7160*K1**2 - 256*K1*K2**3*K3*K4 + 3520*K1*K2**3*K3 + 832*K1*K2**2*K3*K4 - 2496*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 928*K1*K2**2*K5 + 32*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 11376*K1*K2*K3 + 1360*K1*K3*K4 + 128*K1*K4*K5 + 16*K1*K5*K6 - 96*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1056*K2**4*K4 - 4064*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 1952*K2**2*K3**2 - 648*K2**2*K4**2 + 2968*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 4176*K2**2 + 840*K2*K3*K5 + 120*K2*K4*K6 + 8*K3**2*K6 - 2644*K3**2 - 736*K4**2 - 116*K5**2 - 24*K6**2 + 5806
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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