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Flat knot 6.1004

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-2,0,1,2,3,1,0,2,1,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1004']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 4*K1*K2 - 4*K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1004', '6.1690']
Outer characteristic polynomial of the knot is: t^7+41t^5+46t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1004']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 704*K1**4*K2 - 1088*K1**4 + 32*K1**3*K2*K3 - 32*K1**3*K3 - 256*K1**2*K2**4 + 1568*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 6064*K1**2*K2**2 - 224*K1**2*K2*K4 + 6744*K1**2*K2 - 64*K1**2*K3**2 - 16*K1**2*K4**2 - 4556*K1**2 + 864*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5056*K1*K2*K3 + 440*K1*K3*K4 + 16*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 1592*K2**4 - 448*K2**2*K3**2 - 48*K2**2*K4**2 + 1208*K2**2*K4 - 2616*K2**2 + 152*K2*K3*K5 - 1264*K3**2 - 322*K4**2 - 28*K5**2 + 3304
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1004']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4755', 'vk6.5084', 'vk6.6305', 'vk6.6746', 'vk6.8270', 'vk6.8721', 'vk6.9648', 'vk6.9965', 'vk6.20394', 'vk6.21739', 'vk6.27728', 'vk6.29270', 'vk6.39168', 'vk6.41392', 'vk6.45892', 'vk6.47537', 'vk6.48787', 'vk6.49000', 'vk6.49607', 'vk6.49812', 'vk6.50811', 'vk6.51028', 'vk6.51290', 'vk6.51487', 'vk6.57263', 'vk6.58484', 'vk6.61911', 'vk6.63016', 'vk6.66876', 'vk6.67750', 'vk6.69504', 'vk6.70222']
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 O1O2O3O4U2U3O5U4O6U1U6U5
R3 orbit {'O1O2O3O4U2U3O5U4O6U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U4O6U1O5U2U3
Gauss code of K* O1O2O3U1U4U5U6O4O5U3O6U2
Gauss code of -K* O1O2O3U2O4U1O5O6U4U5U6U3
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 -2 0 1 2 1],[ 2 0 -2 0 2 3 1],[ 2 2 0 1 2 1 0],[ 0 0 -1 0 1 1 0],[-1 -2 -2 -1 0 1 0],[-2 -3 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 0 -1 -1 -1 -3],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -2 -2],[ 0 1 0 1 0 -1 0],[ 2 1 0 2 1 0 2],[ 2 3 1 2 0 -2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,0,1,1,1,3,0,0,0,1,1,2,2,1,0,-2]
Phi over symmetry [-2,-2,0,1,1,2,-2,0,1,2,3,1,0,2,1,0,1,1,0,0,1]
Phi of -K [-2,-2,0,1,1,2,-2,1,1,3,3,2,1,2,1,0,1,1,0,0,1]
Phi of K* [-2,-1,-1,0,2,2,0,1,1,1,3,0,0,1,1,1,2,3,2,1,-2]
Phi of -K* [-2,-2,0,1,1,2,-2,0,1,2,3,1,0,2,1,0,1,1,0,0,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+27t^4+15t^2
Outer characteristic polynomial t^7+41t^5+46t^3+4t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 4*K1*K2 - 4*K1 + 5*K2 + 6
2-strand cable arrow polynomial -320*K1**4*K2**2 + 704*K1**4*K2 - 1088*K1**4 + 32*K1**3*K2*K3 - 32*K1**3*K3 - 256*K1**2*K2**4 + 1568*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 6064*K1**2*K2**2 - 224*K1**2*K2*K4 + 6744*K1**2*K2 - 64*K1**2*K3**2 - 16*K1**2*K4**2 - 4556*K1**2 + 864*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5056*K1*K2*K3 + 440*K1*K3*K4 + 16*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 1592*K2**4 - 448*K2**2*K3**2 - 48*K2**2*K4**2 + 1208*K2**2*K4 - 2616*K2**2 + 152*K2*K3*K5 - 1264*K3**2 - 322*K4**2 - 28*K5**2 + 3304
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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