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Flat knot 6.1075

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,0,2,2,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1075']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.697', '6.1075', '6.1524', '6.1733']
Outer characteristic polynomial of the knot is: t^7+34t^5+74t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1075']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 320*K1**4 + 128*K1**3*K2**3*K3 + 736*K1**3*K2*K3 - 320*K1**2*K2**4 + 224*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2832*K1**2*K2**2 + 1720*K1**2*K2 - 1184*K1**2*K3**2 - 32*K1**2*K5**2 - 2176*K1**2 + 704*K1*K2**3*K3 + 192*K1*K2*K3**3 + 4856*K1*K2*K3 + 960*K1*K3*K4 + 88*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 720*K2**4 - 688*K2**2*K3**2 - 64*K2**2*K4**2 + 592*K2**2*K4 - 1818*K2**2 + 352*K2*K3*K5 + 56*K2*K4*K6 - 96*K3**4 + 48*K3**2*K6 - 1880*K3**2 - 504*K4**2 - 144*K5**2 - 62*K6**2 + 2486
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1075']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11443', 'vk6.11739', 'vk6.12754', 'vk6.13098', 'vk6.20326', 'vk6.21669', 'vk6.27626', 'vk6.29172', 'vk6.31192', 'vk6.31531', 'vk6.32356', 'vk6.32771', 'vk6.39050', 'vk6.41312', 'vk6.45802', 'vk6.47479', 'vk6.52208', 'vk6.52471', 'vk6.53036', 'vk6.53357', 'vk6.57185', 'vk6.58398', 'vk6.61795', 'vk6.62918', 'vk6.63772', 'vk6.63883', 'vk6.64197', 'vk6.64384', 'vk6.66794', 'vk6.67664', 'vk6.69430', 'vk6.70154']
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 O1O2O3O4U5U4O6U2O5U1U3U6
R3 orbit {'O1O2O3O4U5U4O6U2O5U1U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U4O6U3O5U1U6
Gauss code of K* O1O2O3U1U4U2U5O6O5U3O4U6
Gauss code of -K* O1O2O3U4O5U1O6O4U6U2U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -1 1 1 -1 2],[ 2 0 1 2 1 0 2],[ 1 -1 0 0 0 0 1],[-1 -2 0 0 1 -2 0],[-1 -1 0 -1 0 -1 -1],[ 1 0 0 2 1 0 2],[-2 -2 -1 0 1 -2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 -1 0 -1 -1],[-1 0 1 0 0 -2 -2],[ 1 1 0 0 0 0 -1],[ 1 2 1 2 0 0 0],[ 2 2 1 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,0,2,2,0,1,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,0,2,2,0,1,0]
Phi of -K [-2,-1,-1,1,1,2,0,1,1,2,2,0,2,2,2,0,1,1,-1,1,2]
Phi of K* [-2,-1,-1,1,1,2,1,2,1,2,2,1,0,2,1,1,2,2,0,1,0]
Phi of -K* [-2,-1,-1,1,1,2,0,1,1,2,2,0,1,2,2,0,0,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial -8w^3z+17w^2z+19w
Inner characteristic polynomial t^6+22t^4+24t^2
Outer characteristic polynomial t^7+34t^5+74t^3
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 96*K1**4*K2 - 320*K1**4 + 128*K1**3*K2**3*K3 + 736*K1**3*K2*K3 - 320*K1**2*K2**4 + 224*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2832*K1**2*K2**2 + 1720*K1**2*K2 - 1184*K1**2*K3**2 - 32*K1**2*K5**2 - 2176*K1**2 + 704*K1*K2**3*K3 + 192*K1*K2*K3**3 + 4856*K1*K2*K3 + 960*K1*K3*K4 + 88*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 720*K2**4 - 688*K2**2*K3**2 - 64*K2**2*K4**2 + 592*K2**2*K4 - 1818*K2**2 + 352*K2*K3*K5 + 56*K2*K4*K6 - 96*K3**4 + 48*K3**2*K6 - 1880*K3**2 - 504*K4**2 - 144*K5**2 - 62*K6**2 + 2486
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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