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Flat knot 6.610

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,0,1,3,5,0,0,1,3,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.610', '7.17787']
Arrow polynomial of the knot is: -16*K1**2 - 4*K1*K2 + 2*K1 + 8*K2 + 2*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.610', '6.1535']
Outer characteristic polynomial of the knot is: t^7+74t^5+97t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.610']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 384*K1**4*K2**2 + 1216*K1**4*K2 - 4864*K1**4 + 448*K1**3*K2*K3 - 704*K1**3*K3 - 7008*K1**2*K2**2 - 384*K1**2*K2*K4 + 11664*K1**2*K2 - 768*K1**2*K3**2 - 64*K1**2*K4**2 - 5776*K1**2 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 + 8880*K1*K2*K3 + 1424*K1*K3*K4 + 144*K1*K4*K5 - 1440*K2**4 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 1600*K2**2*K4 - 4956*K2**2 + 368*K2*K3*K5 + 32*K2*K4*K6 - 2632*K3**2 - 784*K4**2 - 120*K5**2 - 12*K6**2 + 5574
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.610']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11576', 'vk6.11916', 'vk6.12921', 'vk6.13233', 'vk6.20959', 'vk6.22375', 'vk6.28426', 'vk6.31361', 'vk6.31767', 'vk6.32527', 'vk6.32928', 'vk6.40132', 'vk6.42143', 'vk6.46647', 'vk6.52349', 'vk6.52612', 'vk6.53483', 'vk6.58952', 'vk6.64480', 'vk6.69788']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U3O5U2O6U1U6U5U4
R3 orbit {'O1O2O3O4U3O5U2O6U1U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U6U4O6U3O5U2
Gauss code of K* O1O2O3O4U1U5U6U4O6U3O5U2
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -2 -1 3 2 1],[ 3 0 0 0 5 3 1],[ 2 0 0 0 3 1 0],[ 1 0 0 0 1 0 0],[-3 -5 -3 -1 0 0 0],[-2 -3 -1 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 0 0 -1 -3 -5],[-2 0 0 0 0 -1 -3],[-1 0 0 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 2 3 1 0 0 0 0],[ 3 5 3 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,0,0,1,3,5,0,0,1,3,0,0,1,0,0,0]
Phi over symmetry [-3,-2,-1,1,2,3,0,0,1,3,5,0,0,1,3,0,0,1,0,0,0]
Phi of -K [-3,-2,-1,1,2,3,1,2,3,2,1,1,3,3,2,2,3,3,1,2,1]
Phi of K* [-3,-2,-1,1,2,3,1,2,3,2,1,1,3,3,2,2,3,3,1,2,1]
Phi of -K* [-3,-2,-1,1,2,3,0,0,1,3,5,0,0,1,3,0,0,1,0,0,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+22z+37
Enhanced Jones-Krushkal polynomial 2w^3z^2+22w^2z+37w
Inner characteristic polynomial t^6+46t^4+37t^2+1
Outer characteristic polynomial t^7+74t^5+97t^3+7t
Flat arrow polynomial -16*K1**2 - 4*K1*K2 + 2*K1 + 8*K2 + 2*K3 + 9
2-strand cable arrow polynomial -384*K1**6 - 384*K1**4*K2**2 + 1216*K1**4*K2 - 4864*K1**4 + 448*K1**3*K2*K3 - 704*K1**3*K3 - 7008*K1**2*K2**2 - 384*K1**2*K2*K4 + 11664*K1**2*K2 - 768*K1**2*K3**2 - 64*K1**2*K4**2 - 5776*K1**2 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 + 8880*K1*K2*K3 + 1424*K1*K3*K4 + 144*K1*K4*K5 - 1440*K2**4 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 1600*K2**2*K4 - 4956*K2**2 + 368*K2*K3*K5 + 32*K2*K4*K6 - 2632*K3**2 - 784*K4**2 - 120*K5**2 - 12*K6**2 + 5574
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice True
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