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Flat knot 6.1331

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,1,2,3,1,0,1,1,1,2,2,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1331']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.520', '6.682', '6.706', '6.748', '6.1331']
Outer characteristic polynomial of the knot is: t^7+68t^5+35t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1331']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1888*K1**4*K2 - 4704*K1**4 + 384*K1**3*K2*K3 - 448*K1**3*K3 - 192*K1**2*K2**4 + 672*K1**2*K2**3 - 5712*K1**2*K2**2 - 288*K1**2*K2*K4 + 8736*K1**2*K2 - 32*K1**2*K3**2 - 3040*K1**2 + 224*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3880*K1*K2*K3 + 80*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 496*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 512*K2**2*K4 - 2872*K2**2 + 24*K2*K3*K5 - 624*K3**2 - 76*K4**2 + 2938
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1331']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11445', 'vk6.11741', 'vk6.12756', 'vk6.13100', 'vk6.20319', 'vk6.21660', 'vk6.27619', 'vk6.29163', 'vk6.31203', 'vk6.31542', 'vk6.32367', 'vk6.32783', 'vk6.39047', 'vk6.41307', 'vk6.45799', 'vk6.47474', 'vk6.52198', 'vk6.52460', 'vk6.53026', 'vk6.53346', 'vk6.57190', 'vk6.58401', 'vk6.61800', 'vk6.62921', 'vk6.63770', 'vk6.63881', 'vk6.64195', 'vk6.64382', 'vk6.66803', 'vk6.67671', 'vk6.69439', 'vk6.70161']
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 O1O2O3U1O4O5U2U6U4O6U3U5
R3 orbit {'O1O2O3U1O4O5U2U6U4O6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O5U6U5U2O4O6U3
Gauss code of K* O1O2O3U2O4O5U6U1U4O6U3U5
Gauss code of -K* O1O2O3U4U1O5U6U3U5O4O6U2
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 1 1 3 -1],[ 2 0 1 2 1 2 2],[ 2 -1 0 2 1 3 1],[-1 -2 -2 0 1 2 -2],[-1 -1 -1 -1 0 0 -1],[-3 -2 -3 -2 0 0 -3],[ 1 -2 -1 2 1 3 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -2 -3 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -2 -2 -2],[ 1 3 1 2 0 -2 -1],[ 2 2 1 2 2 0 1],[ 2 3 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,2,3,2,3,1,1,1,1,2,2,2,2,1,-1]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,1,2,3,1,0,1,1,1,2,2,0,-1,-1]
Phi of -K [-2,-2,-1,1,1,3,-1,-1,1,2,3,0,1,2,2,0,1,1,-1,0,2]
Phi of K* [-3,-1,-1,1,2,2,0,2,1,2,3,1,0,1,1,1,2,2,0,-1,-1]
Phi of -K* [-2,-2,-1,1,1,3,-1,1,1,2,3,2,1,2,2,1,2,3,-1,0,2]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+48t^4+19t^2+1
Outer characteristic polynomial t^7+68t^5+35t^3+4t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1888*K1**4*K2 - 4704*K1**4 + 384*K1**3*K2*K3 - 448*K1**3*K3 - 192*K1**2*K2**4 + 672*K1**2*K2**3 - 5712*K1**2*K2**2 - 288*K1**2*K2*K4 + 8736*K1**2*K2 - 32*K1**2*K3**2 - 3040*K1**2 + 224*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3880*K1*K2*K3 + 80*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 496*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 512*K2**2*K4 - 2872*K2**2 + 24*K2*K3*K5 - 624*K3**2 - 76*K4**2 + 2938
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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