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Flat knot 6.469

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,0,2,4,4,3,1,0,1,1,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.469']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.469']
Outer characteristic polynomial of the knot is: t^7+75t^5+132t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.469']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 256*K1**4*K2**3 - 960*K1**4*K2**2 + 1664*K1**4*K2 - 3248*K1**4 + 544*K1**3*K2*K3 - 640*K1**3*K3 - 1344*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3264*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8752*K1**2*K2**2 - 832*K1**2*K2*K4 + 8672*K1**2*K2 - 496*K1**2*K3**2 - 48*K1**2*K4**2 - 4512*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2464*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7544*K1*K2*K3 + 1168*K1*K3*K4 + 160*K1*K4*K5 + 24*K1*K5*K6 - 224*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 544*K2**4*K4 - 2624*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1408*K2**2*K3**2 - 384*K2**2*K4**2 + 1616*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 2482*K2**2 - 32*K2*K3**2*K4 + 672*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 2000*K3**2 - 646*K4**2 - 176*K5**2 - 30*K6**2 + 4156
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.469']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4695', 'vk6.5000', 'vk6.6185', 'vk6.6658', 'vk6.8182', 'vk6.8602', 'vk6.9564', 'vk6.9905', 'vk6.17389', 'vk6.20922', 'vk6.20973', 'vk6.22332', 'vk6.22397', 'vk6.23554', 'vk6.23893', 'vk6.28402', 'vk6.36149', 'vk6.40060', 'vk6.40166', 'vk6.42111', 'vk6.43058', 'vk6.43364', 'vk6.46592', 'vk6.46675', 'vk6.48735', 'vk6.49535', 'vk6.49740', 'vk6.51437', 'vk6.55539', 'vk6.58920', 'vk6.65285', 'vk6.69770']
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 O1O2O3O4O5U6U4O6U2U3U1U5
R3 orbit {'O1O2O3O4O5U6U4O6U2U3U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U5U3U4O6U2U6
Gauss code of K* O1O2O3O4U3U1U2U5U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U6U3U4U2
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 -1 -2 0 0 4 -1],[ 1 0 -1 1 1 4 0],[ 2 1 0 1 1 3 1],[ 0 -1 -1 0 1 2 -1],[ 0 -1 -1 -1 0 0 0],[-4 -4 -3 -2 0 0 -4],[ 1 0 -1 1 0 4 0]]
Primitive based matrix [[ 0 4 0 0 -1 -1 -2],[-4 0 0 -2 -4 -4 -3],[ 0 0 0 -1 0 -1 -1],[ 0 2 1 0 -1 -1 -1],[ 1 4 0 1 0 0 -1],[ 1 4 1 1 0 0 -1],[ 2 3 1 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,0,0,1,1,2,0,2,4,4,3,1,0,1,1,1,1,1,0,1,1]
Phi over symmetry [-4,0,0,1,1,2,0,2,4,4,3,1,0,1,1,1,1,1,0,1,1]
Phi of -K [-2,-1,-1,0,0,4,0,0,1,1,3,0,0,0,1,0,1,1,-1,2,4]
Phi of K* [-4,0,0,1,1,2,2,4,1,1,3,1,0,0,1,0,1,1,0,0,0]
Phi of -K* [-2,-1,-1,0,0,4,1,1,1,1,3,0,0,1,4,1,1,4,-1,0,2]
Symmetry type of based matrix c
u-polynomial -t^4+t^2+2t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2-4w^3z+24w^2z+29w
Inner characteristic polynomial t^6+53t^4+63t^2+1
Outer characteristic polynomial t^7+75t^5+132t^3+9t
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -256*K1**6 + 256*K1**4*K2**3 - 960*K1**4*K2**2 + 1664*K1**4*K2 - 3248*K1**4 + 544*K1**3*K2*K3 - 640*K1**3*K3 - 1344*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3264*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8752*K1**2*K2**2 - 832*K1**2*K2*K4 + 8672*K1**2*K2 - 496*K1**2*K3**2 - 48*K1**2*K4**2 - 4512*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2464*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7544*K1*K2*K3 + 1168*K1*K3*K4 + 160*K1*K4*K5 + 24*K1*K5*K6 - 224*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 544*K2**4*K4 - 2624*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1408*K2**2*K3**2 - 384*K2**2*K4**2 + 1616*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 2482*K2**2 - 32*K2*K3**2*K4 + 672*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 2000*K3**2 - 646*K4**2 - 176*K5**2 - 30*K6**2 + 4156
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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