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Flat knot 6.1079

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,2,0,1,1,1,1,1,1,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1079']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.330', '6.531', '6.1076', '6.1079', '6.1567']
Outer characteristic polynomial of the knot is: t^7+31t^5+64t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1079']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 + 640*K1**2*K2**5 - 1984*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3712*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 7168*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 448*K1**2*K2*K4 + 6536*K1**2*K2 - 160*K1**2*K3**2 - 4484*K1**2 + 1568*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 224*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5272*K1*K2*K3 + 632*K1*K3*K4 + 8*K1*K4*K5 + 8*K1*K5*K6 - 736*K2**6 + 352*K2**4*K4 - 2592*K2**4 - 656*K2**2*K3**2 - 160*K2**2*K4**2 + 2160*K2**2*K4 - 1814*K2**2 + 296*K2*K3*K5 + 64*K2*K4*K6 - 1252*K3**2 - 484*K4**2 - 24*K5**2 - 10*K6**2 + 3218
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1079']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11458', 'vk6.11759', 'vk6.12775', 'vk6.13113', 'vk6.20670', 'vk6.22110', 'vk6.28169', 'vk6.29594', 'vk6.31214', 'vk6.31561', 'vk6.32388', 'vk6.32793', 'vk6.39613', 'vk6.41854', 'vk6.46225', 'vk6.47832', 'vk6.52220', 'vk6.52503', 'vk6.53059', 'vk6.53375', 'vk6.57603', 'vk6.58765', 'vk6.62259', 'vk6.63205', 'vk6.63785', 'vk6.63901', 'vk6.64216', 'vk6.64397', 'vk6.67059', 'vk6.67927', 'vk6.69678', 'vk6.70361']
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 O1O2O3O4U5U4O6U3O5U1U2U6
R3 orbit {'O1O2O3O4U5U4O6U3O5U1U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U4O6U2O5U1U6
Gauss code of K* O1O2O3U1U2U4U5O6O5U3O4U6
Gauss code of -K* O1O2O3U4O5U1O6O4U6U5U2U3
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 -2 0 0 1 -1 2],[ 2 0 1 1 1 0 2],[ 0 -1 0 1 1 -2 1],[ 0 -1 -1 0 0 -1 0],[-1 -1 -1 0 0 -1 -1],[ 1 0 2 1 1 0 2],[-2 -2 -1 0 1 -2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 0 -1 -1 -1],[ 0 0 0 0 -1 -1 -1],[ 0 1 1 1 0 -2 -1],[ 1 2 1 1 2 0 0],[ 2 2 1 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,-1,0,1,2,2,0,1,1,1,1,1,1,2,1,0]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,2,2,0,1,1,1,1,1,1,2,1,0]
Phi of -K [-2,-1,0,0,1,2,1,1,1,2,2,-1,0,1,1,-1,0,1,1,2,2]
Phi of K* [-2,-1,0,0,1,2,2,1,2,1,2,0,1,1,2,1,-1,1,0,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,1,1,2,1,2,1,2,-1,0,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial t^6+21t^4+12t^2+1
Outer characteristic polynomial t^7+31t^5+64t^3+9t
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 448*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 + 640*K1**2*K2**5 - 1984*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3712*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 7168*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 448*K1**2*K2*K4 + 6536*K1**2*K2 - 160*K1**2*K3**2 - 4484*K1**2 + 1568*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 224*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5272*K1*K2*K3 + 632*K1*K3*K4 + 8*K1*K4*K5 + 8*K1*K5*K6 - 736*K2**6 + 352*K2**4*K4 - 2592*K2**4 - 656*K2**2*K3**2 - 160*K2**2*K4**2 + 2160*K2**2*K4 - 1814*K2**2 + 296*K2*K3*K5 + 64*K2*K4*K6 - 1252*K3**2 - 484*K4**2 - 24*K5**2 - 10*K6**2 + 3218
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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