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Flat knot 6.759

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,3,2,3,0,2,1,1,2,2,2,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.759']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370']
Outer characteristic polynomial of the knot is: t^7+66t^5+91t^3+24t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.759']
2-strand cable arrow polynomial of the knot is: -576*K1**2*K2**4 + 832*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4016*K1**2*K2**2 - 160*K1**2*K2*K4 + 4456*K1**2*K2 - 64*K1**2*K4**2 - 3432*K1**2 + 608*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 288*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4232*K1*K2*K3 + 464*K1*K3*K4 + 120*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 1920*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 2080*K2**2*K4 - 2072*K2**2 + 144*K2*K3*K5 - 1136*K3**2 - 528*K4**2 - 48*K5**2 + 2574
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.759']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17540', 'vk6.17545', 'vk6.17597', 'vk6.17600', 'vk6.24048', 'vk6.24053', 'vk6.24140', 'vk6.24143', 'vk6.36324', 'vk6.36333', 'vk6.36395', 'vk6.36398', 'vk6.43449', 'vk6.43454', 'vk6.43496', 'vk6.43498', 'vk6.55630', 'vk6.55643', 'vk6.55657', 'vk6.55666', 'vk6.60148', 'vk6.60158', 'vk6.60205', 'vk6.60216', 'vk6.65331', 'vk6.65346', 'vk6.65366', 'vk6.65376', 'vk6.68501', 'vk6.68510', 'vk6.68525', 'vk6.68532']
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 O1O2O3O4U2O5U6U4U3O6U1U5
R3 orbit {'O1O2O3O4U2O5U6U4U3O6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4O6U2U1U6O5U3
Gauss code of K* O1O2O3U1O4O5U4U6U3U2O6U5
Gauss code of -K* O1O2O3U4O5U2U1U5U6O4O6U3
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 -1 -2 1 1 3 -2],[ 1 0 -2 2 2 3 -1],[ 2 2 0 2 1 2 0],[-1 -2 -2 0 0 1 -2],[-1 -2 -1 0 0 0 -1],[-3 -3 -2 -1 0 0 -3],[ 2 1 0 2 1 3 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -1 -3 -2 -3],[-1 0 0 0 -2 -1 -1],[-1 1 0 0 -2 -2 -2],[ 1 3 2 2 0 -2 -1],[ 2 2 1 2 2 0 0],[ 2 3 1 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,1,3,2,3,0,2,1,1,2,2,2,2,1,0]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,3,2,3,0,2,1,1,2,2,2,2,1,0]
Phi of -K [-2,-2,-1,1,1,3,0,-1,1,2,3,0,1,2,2,0,0,1,0,1,2]
Phi of K* [-3,-1,-1,1,2,2,1,2,1,2,3,0,0,1,1,0,2,2,0,-1,0]
Phi of -K* [-2,-2,-1,1,1,3,0,1,1,2,3,2,1,2,2,2,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial -4w^4z^2+10w^3z^2-8w^3z+27w^2z+15w
Inner characteristic polynomial t^6+46t^4+51t^2+9
Outer characteristic polynomial t^7+66t^5+91t^3+24t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial -576*K1**2*K2**4 + 832*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4016*K1**2*K2**2 - 160*K1**2*K2*K4 + 4456*K1**2*K2 - 64*K1**2*K4**2 - 3432*K1**2 + 608*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 288*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4232*K1*K2*K3 + 464*K1*K3*K4 + 120*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 1920*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 2080*K2**2*K4 - 2072*K2**2 + 144*K2*K3*K5 - 1136*K3**2 - 528*K4**2 - 48*K5**2 + 2574
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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