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Flat knot 6.999

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,-1,1,2,2,3,1,1,0,0,1,1,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.999']
Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182']
Outer characteristic polynomial of the knot is: t^7+47t^5+63t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.999']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 704*K1**4*K2 - 1616*K1**4 + 480*K1**3*K2*K3 - 480*K1**3*K3 + 288*K1**2*K2**3 - 4432*K1**2*K2**2 - 448*K1**2*K2*K4 + 5784*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 3864*K1**2 + 128*K1*K2**3*K3 - 512*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 32*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5848*K1*K2*K3 + 1080*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 368*K2**4 - 368*K2**2*K3**2 - 56*K2**2*K4**2 + 656*K2**2*K4 - 2998*K2**2 + 280*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1716*K3**2 - 416*K4**2 - 68*K5**2 - 10*K6**2 + 3150
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.999']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4656', 'vk6.4943', 'vk6.6106', 'vk6.6593', 'vk6.8119', 'vk6.8521', 'vk6.9509', 'vk6.9864', 'vk6.20376', 'vk6.21719', 'vk6.27692', 'vk6.29238', 'vk6.39132', 'vk6.41388', 'vk6.45872', 'vk6.47535', 'vk6.48696', 'vk6.48899', 'vk6.49456', 'vk6.49675', 'vk6.50716', 'vk6.50915', 'vk6.51199', 'vk6.51400', 'vk6.57237', 'vk6.58464', 'vk6.61859', 'vk6.62996', 'vk6.66860', 'vk6.67730', 'vk6.69488', 'vk6.70212']
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 O1O2O3O4U2U1O5U3O6U5U6U4
R3 orbit {'O1O2O3O4U2U1O5U3O6U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U6O5U2O6U4U3
Gauss code of K* O1O2O3U4U5U6U3O5O4U1O6U2
Gauss code of -K* O1O2O3U2O4U3O5O6U1U4U6U5
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 3 0 1],[ 2 0 0 2 3 1 0],[ 2 0 0 1 2 1 0],[ 0 -2 -1 0 2 1 1],[-3 -3 -2 -2 0 -1 1],[ 0 -1 -1 -1 1 0 1],[-1 0 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 1 -1 -2 -2 -3],[-1 -1 0 -1 -1 0 0],[ 0 1 1 0 -1 -1 -1],[ 0 2 1 1 0 -1 -2],[ 2 2 0 1 1 0 0],[ 2 3 0 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,-1,1,2,2,3,1,1,0,0,1,1,1,1,2,0]
Phi over symmetry [-3,-1,0,0,2,2,-1,1,2,2,3,1,1,0,0,1,1,1,1,2,0]
Phi of -K [-2,-2,0,0,1,3,0,0,1,3,2,1,1,3,3,-1,0,1,0,2,3]
Phi of K* [-3,-1,0,0,2,2,3,1,2,2,3,0,0,3,3,1,0,1,1,1,0]
Phi of -K* [-2,-2,0,0,1,3,0,1,1,0,2,1,2,0,3,-1,1,1,1,2,-1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+29t^4+21t^2+1
Outer characteristic polynomial t^7+47t^5+63t^3+6t
Flat arrow polynomial -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 704*K1**4*K2 - 1616*K1**4 + 480*K1**3*K2*K3 - 480*K1**3*K3 + 288*K1**2*K2**3 - 4432*K1**2*K2**2 - 448*K1**2*K2*K4 + 5784*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 3864*K1**2 + 128*K1*K2**3*K3 - 512*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 32*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5848*K1*K2*K3 + 1080*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 368*K2**4 - 368*K2**2*K3**2 - 56*K2**2*K4**2 + 656*K2**2*K4 - 2998*K2**2 + 280*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1716*K3**2 - 416*K4**2 - 68*K5**2 - 10*K6**2 + 3150
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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