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Flat knot 6.973

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,1,1,1,0,-1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.973']
Arrow polynomial of the knot is: -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.574', '6.593', '6.604', '6.649', '6.650', '6.673', '6.690', '6.783', '6.973', '6.985', '6.1033', '6.1035']
Outer characteristic polynomial of the knot is: t^7+56t^5+77t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.973']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 384*K1**4*K2**2 + 704*K1**4*K2 - 3488*K1**4 + 256*K1**3*K2*K3 - 544*K1**3*K3 + 704*K1**2*K2**3 - 7520*K1**2*K2**2 - 384*K1**2*K2*K4 + 9992*K1**2*K2 - 352*K1**2*K3**2 - 4500*K1**2 + 416*K1*K2**3*K3 - 736*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7544*K1*K2*K3 + 608*K1*K3*K4 + 24*K1*K4*K5 - 2488*K2**4 - 624*K2**2*K3**2 - 8*K2**2*K4**2 + 2112*K2**2*K4 - 3302*K2**2 + 416*K2*K3*K5 + 8*K2*K4*K6 - 1744*K3**2 - 482*K4**2 - 68*K5**2 - 2*K6**2 + 4160
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.973']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17017', 'vk6.17259', 'vk6.20233', 'vk6.21529', 'vk6.23433', 'vk6.23735', 'vk6.27446', 'vk6.29052', 'vk6.35510', 'vk6.35959', 'vk6.38861', 'vk6.41049', 'vk6.42930', 'vk6.43227', 'vk6.45614', 'vk6.47369', 'vk6.55197', 'vk6.55434', 'vk6.57069', 'vk6.58207', 'vk6.59584', 'vk6.59908', 'vk6.61600', 'vk6.62778', 'vk6.65000', 'vk6.65205', 'vk6.66694', 'vk6.67541', 'vk6.68280', 'vk6.68432', 'vk6.69345', 'vk6.70093']
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 O1O2O3O4U1U2O5U4O6U3U6U5
R3 orbit {'O1O2O3O4U1U2O5U4O6U3U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U2O6U1O5U3U4
Gauss code of K* O1O2O3U4U5U1U6O4O5U3O6U2
Gauss code of -K* O1O2O3U2O4U1O5O6U4U3U5U6
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 -3 -1 0 1 2 1],[ 3 0 1 3 2 2 1],[ 1 -1 0 2 1 2 1],[ 0 -3 -2 0 0 3 1],[-1 -2 -1 0 0 1 0],[-2 -2 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -3 -2 -2],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 0 -1 -2],[ 0 3 1 0 0 -2 -3],[ 1 2 1 1 2 0 -1],[ 3 2 1 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,3,2,2,0,1,1,1,0,1,2,2,3,1]
Phi over symmetry [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,1,1,1,0,-1,0,0,1]
Phi of -K [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,1,1,1,0,-1,0,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,-1,1,3,0,1,1,2,0,1,3,-1,0,1]
Phi of -K* [-3,-1,0,1,1,2,1,3,1,2,2,2,1,1,2,1,0,3,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+40t^4+24t^2+1
Outer characteristic polynomial t^7+56t^5+77t^3+10t
Flat arrow polynomial -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -128*K1**6 - 384*K1**4*K2**2 + 704*K1**4*K2 - 3488*K1**4 + 256*K1**3*K2*K3 - 544*K1**3*K3 + 704*K1**2*K2**3 - 7520*K1**2*K2**2 - 384*K1**2*K2*K4 + 9992*K1**2*K2 - 352*K1**2*K3**2 - 4500*K1**2 + 416*K1*K2**3*K3 - 736*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7544*K1*K2*K3 + 608*K1*K3*K4 + 24*K1*K4*K5 - 2488*K2**4 - 624*K2**2*K3**2 - 8*K2**2*K4**2 + 2112*K2**2*K4 - 3302*K2**2 + 416*K2*K3*K5 + 8*K2*K4*K6 - 1744*K3**2 - 482*K4**2 - 68*K5**2 - 2*K6**2 + 4160
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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