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Flat knot 6.778

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,2,1,2,1,1,0,1,0,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.778']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^7+27t^5+27t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.778']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 192*K1**4*K2**2 + 2048*K1**4*K2 - 6736*K1**4 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2048*K1**3*K3 - 4704*K1**2*K2**2 - 1056*K1**2*K2*K4 + 13368*K1**2*K2 - 688*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 7152*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 8560*K1*K2*K3 + 1896*K1*K3*K4 + 184*K1*K4*K5 - 240*K2**4 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 1024*K2**2*K4 - 6300*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 - 2932*K3**2 - 916*K4**2 - 124*K5**2 - 12*K6**2 + 6426
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.778']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3644', 'vk6.3739', 'vk6.3934', 'vk6.4029', 'vk6.4487', 'vk6.4584', 'vk6.5873', 'vk6.6002', 'vk6.7127', 'vk6.7306', 'vk6.7397', 'vk6.7926', 'vk6.8047', 'vk6.9360', 'vk6.17928', 'vk6.18025', 'vk6.18740', 'vk6.24467', 'vk6.24863', 'vk6.25326', 'vk6.37487', 'vk6.43894', 'vk6.44218', 'vk6.44523', 'vk6.48284', 'vk6.48347', 'vk6.50065', 'vk6.50179', 'vk6.50587', 'vk6.50652', 'vk6.55873', 'vk6.60718']
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 O1O2O3O4U3O5U4U1U5O6U2U6
R3 orbit {'O1O2O3O4U3O5U4U1U5O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3O5U6U4U1O6U2
Gauss code of K* O1O2O3U4O5O4U2U5U6U1O6U3
Gauss code of -K* O1O2O3U1O4U3U4U5U2O6O5U6
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 0 -1 0 2 1],[ 2 0 2 -1 1 2 1],[ 0 -2 0 -1 0 1 1],[ 1 1 1 0 1 1 0],[ 0 -1 0 -1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -1 -2],[-1 0 0 0 -1 0 -1],[ 0 1 0 0 0 -1 -1],[ 0 1 1 0 0 -1 -2],[ 1 1 0 1 1 0 1],[ 2 2 1 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,1,1,2,0,1,0,1,0,1,1,1,2,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,2,1,2,1,1,0,1,0,0,1,1,1,0]
Phi of -K [-2,-1,0,0,1,2,2,0,1,2,2,0,0,2,2,0,0,1,1,1,1]
Phi of K* [-2,-1,0,0,1,2,1,1,1,2,2,0,1,2,2,0,0,0,0,1,2]
Phi of -K* [-2,-1,0,0,1,2,-1,1,2,1,2,1,1,0,1,0,0,1,1,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+17t^4+11t^2
Outer characteristic polynomial t^7+27t^5+27t^3+4t
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -384*K1**6 - 192*K1**4*K2**2 + 2048*K1**4*K2 - 6736*K1**4 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2048*K1**3*K3 - 4704*K1**2*K2**2 - 1056*K1**2*K2*K4 + 13368*K1**2*K2 - 688*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 7152*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 8560*K1*K2*K3 + 1896*K1*K3*K4 + 184*K1*K4*K5 - 240*K2**4 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 1024*K2**2*K4 - 6300*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 - 2932*K3**2 - 916*K4**2 - 124*K5**2 - 12*K6**2 + 6426
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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