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Flat knot 6.199

Min(phi) over symmetries of the knot is: [-3,-3,-2,2,3,3,-1,0,3,3,4,0,2,1,3,3,2,3,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.199']
Arrow polynomial of the knot is: 16*K1**3 - 8*K1**2 - 8*K1*K2 - 8*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.199', '6.890']
Outer characteristic polynomial of the knot is: t^7+124t^5+176t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.199']
2-strand cable arrow polynomial of the knot is: -736*K1**4 - 64*K1**3*K3 - 512*K1**2*K2**4 + 1728*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7456*K1**2*K2**2 - 512*K1**2*K2*K4 + 10112*K1**2*K2 - 32*K1**2*K3**2 - 64*K1**2*K4**2 - 8184*K1**2 + 960*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 - 256*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9344*K1*K2*K3 + 976*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6 + 128*K2**4*K4 - 3072*K2**4 - 1152*K2**2*K3**2 - 192*K2**2*K4**2 + 3216*K2**2*K4 - 5288*K2**2 - 64*K2*K3**2*K4 + 752*K2*K3*K5 + 160*K2*K4*K6 - 2776*K3**2 - 912*K4**2 - 160*K5**2 - 32*K6**2 + 6118
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.199']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81590', 'vk6.81676', 'vk6.81693', 'vk6.81757', 'vk6.81772', 'vk6.81974', 'vk6.82279', 'vk6.82413', 'vk6.82446', 'vk6.82524', 'vk6.82704', 'vk6.83210', 'vk6.83617', 'vk6.84189', 'vk6.84396', 'vk6.85988', 'vk6.88180', 'vk6.88756', 'vk6.88786', 'vk6.89113']
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 O1O2O3O4O5U2O6U3U1U5U6U4
R3 orbit {'O1O2O3O4O5U2O6U3U1U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U1U5U3O6U4
Gauss code of K* O1O2O3O4O5U2U6U1U5U3O6U4
Gauss code of -K* Same
Diagrammatic symmetry type -
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 -3 -2 3 2 3],[ 3 0 -1 1 5 3 3],[ 3 1 0 1 3 2 2],[ 2 -1 -1 0 3 1 2],[-3 -5 -3 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 2 -2 -3 -3],[-3 0 1 -1 -3 -3 -5],[-3 -1 0 -1 -2 -2 -3],[-2 1 1 0 -1 -2 -3],[ 2 3 2 1 0 -1 -1],[ 3 3 2 2 1 0 1],[ 3 5 3 3 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-2,2,3,3,-1,1,3,3,5,1,2,2,3,1,2,3,1,1,-1]
Phi over symmetry [-3,-3,-2,2,3,3,-1,0,3,3,4,0,2,1,3,3,2,3,0,0,-1]
Phi of -K [-3,-3,-2,2,3,3,-1,0,3,3,4,0,2,1,3,3,2,3,0,0,-1]
Phi of K* [-3,-3,-2,2,3,3,-1,0,3,3,4,0,2,1,3,3,2,3,0,0,-1]
Phi of -K* [-3,-3,-2,2,3,3,-1,1,3,3,5,1,2,2,3,1,2,3,1,1,-1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+26z+29
Enhanced Jones-Krushkal polynomial 6w^3z^2+26w^2z+29w
Inner characteristic polynomial t^6+80t^4+28t^2+1
Outer characteristic polynomial t^7+124t^5+176t^3+7t
Flat arrow polynomial 16*K1**3 - 8*K1**2 - 8*K1*K2 - 8*K1 + 4*K2 + 5
2-strand cable arrow polynomial -736*K1**4 - 64*K1**3*K3 - 512*K1**2*K2**4 + 1728*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7456*K1**2*K2**2 - 512*K1**2*K2*K4 + 10112*K1**2*K2 - 32*K1**2*K3**2 - 64*K1**2*K4**2 - 8184*K1**2 + 960*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 - 256*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9344*K1*K2*K3 + 976*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6 + 128*K2**4*K4 - 3072*K2**4 - 1152*K2**2*K3**2 - 192*K2**2*K4**2 + 3216*K2**2*K4 - 5288*K2**2 - 64*K2*K3**2*K4 + 752*K2*K3*K5 + 160*K2*K4*K6 - 2776*K3**2 - 912*K4**2 - 160*K5**2 - 32*K6**2 + 6118
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice True
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