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Flat knot 6.1697

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,0,0,1,1,-1,-1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.1697']
Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.313', '6.623', '6.1031', '6.1201', '6.1327', '6.1378', '6.1640', '6.1697', '6.1797', '6.1833']
Outer characteristic polynomial of the knot is: t^7+24t^5+64t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1697']
2-strand cable arrow polynomial of the knot is: -32*K1**4 - 1984*K1**2*K2**4 + 2720*K1**2*K2**3 - 6528*K1**2*K2**2 - 256*K1**2*K2*K4 + 4848*K1**2*K2 - 3108*K1**2 + 1984*K1*K2**3*K3 - 832*K1*K2**2*K3 - 64*K1*K2**2*K5 + 4384*K1*K2*K3 + 208*K1*K3*K4 - 448*K2**6 + 544*K2**4*K4 - 2264*K2**4 - 528*K2**2*K3**2 - 208*K2**2*K4**2 + 1400*K2**2*K4 - 936*K2**2 + 48*K2*K3*K5 + 24*K2*K4*K6 - 860*K3**2 - 302*K4**2 + 2188
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1697']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10504', 'vk6.10508', 'vk6.10569', 'vk6.10577', 'vk6.10758', 'vk6.10766', 'vk6.10881', 'vk6.10885', 'vk6.17684', 'vk6.17686', 'vk6.17733', 'vk6.17735', 'vk6.24294', 'vk6.24296', 'vk6.30189', 'vk6.30193', 'vk6.30256', 'vk6.30264', 'vk6.30385', 'vk6.30393', 'vk6.36522', 'vk6.36524', 'vk6.43628', 'vk6.43630', 'vk6.43734', 'vk6.43736', 'vk6.60366', 'vk6.60368', 'vk6.63449', 'vk6.63453', 'vk6.65425', 'vk6.65427']
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 O1O2O3U1U2O4O5U4U3O6U5U6
R3 orbit {'O1O2O3U1U2O4O5U4U3O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1U6O5O6U2U3
Gauss code of K* O1O2U3O4O3U5U6U2O5O6U1U4
Gauss code of -K* O1O2U1O3O4U2U4O5O6U3U5U6
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 -1 1 1],[ 2 0 1 2 0 1 0],[ 0 -1 0 1 0 1 0],[-1 -2 -1 0 0 2 1],[ 1 0 0 0 0 1 1],[-1 -1 -1 -2 -1 0 1],[-1 0 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 2 1 -1 0 -2],[-1 -2 0 1 -1 -1 -1],[-1 -1 -1 0 0 -1 0],[ 0 1 1 0 0 0 -1],[ 1 0 1 1 0 0 0],[ 2 2 1 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-2,-1,1,0,2,-1,1,1,1,0,1,0,0,1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,0,0,1,1,-1,-1,-2]
Phi of -K [-2,-1,0,1,1,1,1,1,1,2,3,1,2,1,1,0,0,1,-2,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-2,1,0,1,2,1,0,2,1,1,1,3,1,1,1]
Phi of -K* [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,0,0,1,1,-1,-1,-2]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+18z+17
Enhanced Jones-Krushkal polynomial -4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w
Inner characteristic polynomial t^6+16t^4+19t^2
Outer characteristic polynomial t^7+24t^5+64t^3+7t
Flat arrow polynomial 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
2-strand cable arrow polynomial -32*K1**4 - 1984*K1**2*K2**4 + 2720*K1**2*K2**3 - 6528*K1**2*K2**2 - 256*K1**2*K2*K4 + 4848*K1**2*K2 - 3108*K1**2 + 1984*K1*K2**3*K3 - 832*K1*K2**2*K3 - 64*K1*K2**2*K5 + 4384*K1*K2*K3 + 208*K1*K3*K4 - 448*K2**6 + 544*K2**4*K4 - 2264*K2**4 - 528*K2**2*K3**2 - 208*K2**2*K4**2 + 1400*K2**2*K4 - 936*K2**2 + 48*K2*K3*K5 + 24*K2*K4*K6 - 860*K3**2 - 302*K4**2 + 2188
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
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