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Flat knot 6.244

Min(phi) over symmetries of the knot is: [-4,-1,0,0,1,4,0,2,3,1,5,0,0,0,1,0,0,2,0,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.244']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.84', '6.123', '6.244', '6.864']
Outer characteristic polynomial of the knot is: t^7+87t^5+83t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.244']
2-strand cable arrow polynomial of the knot is: -416*K1**4 + 192*K1**3*K3*K4 + 176*K1**2*K2 - 544*K1**2*K3**2 - 512*K1**2*K4**2 - 888*K1**2 + 768*K1*K2*K3 + 1824*K1*K3*K4 + 544*K1*K4*K5 - 16*K2**2*K4**2 + 128*K2**2*K4 - 460*K2**2 + 80*K2*K3*K5 + 32*K2*K4*K6 + 16*K2*K5*K7 - 32*K3**2*K4**2 - 1024*K3**2 + 48*K3*K4*K7 + 16*K3*K5*K8 - 16*K4**4 + 16*K4**2*K8 - 944*K4**2 - 224*K5**2 - 12*K6**2 - 24*K7**2 - 12*K8**2 + 1282
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.244']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14105', 'vk6.14314', 'vk6.15543', 'vk6.16027', 'vk6.16442', 'vk6.16458', 'vk6.22853', 'vk6.34055', 'vk6.34110', 'vk6.34447', 'vk6.34491', 'vk6.34801', 'vk6.34830', 'vk6.42424', 'vk6.54078', 'vk6.54304', 'vk6.54679', 'vk6.54705', 'vk6.64532', 'vk6.64742']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4O5U4O6U1U6U3U2U5
R3 orbit {'O1O2O3O4O5U4O6U1U6U3U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U4U3U6U5O6U2
Gauss code of K* O1O2O3O4O5U1U4U3U6U5O6U2
Gauss code of -K* Same
Diagrammatic symmetry type -
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 4 1],[ 4 0 3 2 0 5 1],[ 0 -3 0 0 0 3 0],[ 0 -2 0 0 0 2 0],[ 1 0 0 0 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 1 0 0 -1 -4],[-4 0 0 -2 -3 -1 -5],[-1 0 0 0 0 0 -1],[ 0 2 0 0 0 0 -2],[ 0 3 0 0 0 0 -3],[ 1 1 0 0 0 0 0],[ 4 5 1 2 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,0,1,4,0,2,3,1,5,0,0,0,1,0,0,2,0,3,0]
Phi over symmetry [-4,-1,0,0,1,4,0,2,3,1,5,0,0,0,1,0,0,2,0,3,0]
Phi of -K [-4,-1,0,0,1,4,3,1,2,4,3,1,1,2,4,0,1,1,1,2,3]
Phi of K* [-4,-1,0,0,1,4,3,1,2,4,3,1,1,2,4,0,1,1,1,2,3]
Phi of -K* [-4,-1,0,0,1,4,0,2,3,1,5,0,0,0,1,0,0,2,0,3,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -8w^3z+15w^2z+15w
Inner characteristic polynomial t^6+53t^4+27t^2
Outer characteristic polynomial t^7+87t^5+83t^3
Flat arrow polynomial -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
2-strand cable arrow polynomial -416*K1**4 + 192*K1**3*K3*K4 + 176*K1**2*K2 - 544*K1**2*K3**2 - 512*K1**2*K4**2 - 888*K1**2 + 768*K1*K2*K3 + 1824*K1*K3*K4 + 544*K1*K4*K5 - 16*K2**2*K4**2 + 128*K2**2*K4 - 460*K2**2 + 80*K2*K3*K5 + 32*K2*K4*K6 + 16*K2*K5*K7 - 32*K3**2*K4**2 - 1024*K3**2 + 48*K3*K4*K7 + 16*K3*K5*K8 - 16*K4**4 + 16*K4**2*K8 - 944*K4**2 - 224*K5**2 - 12*K6**2 - 24*K7**2 - 12*K8**2 + 1282
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}]]
If K is slice True
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