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Flat knot 6.494

Min(phi) over symmetries of the knot is: [-4,0,0,0,2,2,0,1,3,4,4,0,1,0,1,1,1,1,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.494']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K2**2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.92', '6.348', '6.490', '6.494']
Outer characteristic polynomial of the knot is: t^7+68t^5+135t^3+25t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.494']
2-strand cable arrow polynomial of the knot is: -288*K2**4*K4**2 + 1728*K2**4*K4 - 4544*K2**4 + 96*K2**3*K4*K6 - 256*K2**3*K6 + 64*K2**2*K4**3 - 1360*K2**2*K4**2 + 3784*K2**2*K4 + 340*K2**2 + 456*K2*K4*K6 - 8*K4**4 - 904*K4**2 - 20*K6**2 + 910
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.494']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70469', 'vk6.70486', 'vk6.70532', 'vk6.70610', 'vk6.70645', 'vk6.70671', 'vk6.70764', 'vk6.70847', 'vk6.70925', 'vk6.70956', 'vk6.71013', 'vk6.71121', 'vk6.71159', 'vk6.71176', 'vk6.71253', 'vk6.71307', 'vk6.72393', 'vk6.72410', 'vk6.72745', 'vk6.73057', 'vk6.73610', 'vk6.74398', 'vk6.74927', 'vk6.75390', 'vk6.76493', 'vk6.76685', 'vk6.77738', 'vk6.78351', 'vk6.79446', 'vk6.79950', 'vk6.87182', 'vk6.90127']
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 O1O2O3O4O5U1U4U5O6U3U2U6
R3 orbit {'O1O2O3O4O5U1U4U5O6U3U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U3O6U1U2U5
Gauss code of K* O1O2O3U4O5O6O4U1U6U5U2U3
Gauss code of -K* O1O2O3U1O4O5O6U4U5U3U2U6
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 0 0 0 2 2],[ 4 0 4 3 1 2 2],[ 0 -4 0 0 -1 1 2],[ 0 -3 0 0 -1 1 1],[ 0 -1 1 1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 2 2 0 0 0 -4],[-2 0 0 0 -1 -2 -2],[-2 0 0 -1 -1 -1 -2],[ 0 0 1 0 1 1 -1],[ 0 1 1 -1 0 0 -3],[ 0 2 1 -1 0 0 -4],[ 4 2 2 1 3 4 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,0,4,0,0,1,2,2,1,1,1,2,-1,-1,1,0,3,4]
Phi over symmetry [-4,0,0,0,2,2,0,1,3,4,4,0,1,0,1,1,1,1,2,1,0]
Phi of -K [-4,0,0,0,2,2,0,1,3,4,4,0,1,0,1,1,1,1,2,1,0]
Phi of K* [-2,-2,0,0,0,4,0,0,1,2,4,1,1,1,4,0,-1,0,-1,1,3]
Phi of -K* [-4,0,0,0,2,2,1,3,4,2,2,1,1,0,1,0,1,1,2,1,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 3z^3+17z^2+29z+15
Enhanced Jones-Krushkal polynomial 3w^4z^3+17w^3z^2+29w^2z+15
Inner characteristic polynomial t^6+44t^4+47t^2+1
Outer characteristic polynomial t^7+68t^5+135t^3+25t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K2**2 + 3
2-strand cable arrow polynomial -288*K2**4*K4**2 + 1728*K2**4*K4 - 4544*K2**4 + 96*K2**3*K4*K6 - 256*K2**3*K6 + 64*K2**2*K4**3 - 1360*K2**2*K4**2 + 3784*K2**2*K4 + 340*K2**2 + 456*K2*K4*K6 - 8*K4**4 - 904*K4**2 - 20*K6**2 + 910
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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