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Flat knot 6.1749

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,0,2,2,0,1,0,1,0,1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1749', '7.33028']
Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.474', '6.1684', '6.1716', '6.1749', '6.1781']
Outer characteristic polynomial of the knot is: t^7+34t^5+79t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1749', '7.33028']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 1728*K1**4*K2**2 + 3232*K1**4*K2 - 5552*K1**4 + 2016*K1**3*K2*K3 + 256*K1**3*K3*K4 - 320*K1**3*K3 - 1024*K1**2*K2**4 + 3744*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12064*K1**2*K2**2 - 704*K1**2*K2*K4 + 10024*K1**2*K2 - 1552*K1**2*K3**2 - 32*K1**2*K3*K5 - 384*K1**2*K4**2 - 1772*K1**2 + 2336*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2976*K1*K2**2*K3 - 384*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8720*K1*K2*K3 - 64*K1*K2*K4*K5 + 1624*K1*K3*K4 + 296*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 3080*K2**4 - 32*K2**3*K6 - 1808*K2**2*K3**2 - 320*K2**2*K4**2 + 2448*K2**2*K4 - 2002*K2**2 - 96*K2*K3**2*K4 + 888*K2*K3*K5 + 208*K2*K4*K6 - 1648*K3**2 - 550*K4**2 - 100*K5**2 - 14*K6**2 + 3276
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1749']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.498', 'vk6.590', 'vk6.620', 'vk6.997', 'vk6.1095', 'vk6.1123', 'vk6.1669', 'vk6.1843', 'vk6.2171', 'vk6.2183', 'vk6.2278', 'vk6.2312', 'vk6.2792', 'vk6.2890', 'vk6.3070', 'vk6.3200', 'vk6.5257', 'vk6.6512', 'vk6.8894', 'vk6.9809', 'vk6.20809', 'vk6.21046', 'vk6.22205', 'vk6.22470', 'vk6.28491', 'vk6.29770', 'vk6.39868', 'vk6.40279', 'vk6.46424', 'vk6.46916', 'vk6.49145', 'vk6.58832']
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 O1O2O3U4U5O4O5U1U3O6U2U6
R3 orbit {'O1O2O3U4U5O4O5U1U3O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O4U1U3O5O6U5U6
Gauss code of K* O1O2U3O4O3U1U4U2O5O6U5U6
Gauss code of -K* O1O2U1O3O4U5U6O5O6U3U2U4
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 2 1 1 3 1],[ 0 -2 0 0 -1 1 1],[-1 -1 0 0 -2 0 0],[ 1 -1 1 2 0 1 1],[-1 -3 -1 0 -1 0 1],[-1 -1 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -3],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 -2 -1],[ 0 1 1 0 0 -1 -2],[ 1 1 1 2 1 0 -1],[ 2 3 1 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,1,1,3,0,1,1,1,0,2,1,1,2,1]
Phi over symmetry [-2,-1,0,1,1,1,0,0,0,2,2,0,1,0,1,0,1,0,0,-1,0]
Phi of -K [-2,-1,0,1,1,1,0,0,0,2,2,0,1,0,1,0,1,0,0,-1,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,0,1,2,0,0,1,0,1,0,2,0,0,0]
Phi of -K* [-2,-1,0,1,1,1,1,2,1,1,3,1,1,2,1,1,0,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+26t^4+56t^2+1
Outer characteristic polynomial t^7+34t^5+79t^3+6t
Flat arrow polynomial 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
2-strand cable arrow polynomial -768*K1**6 - 1728*K1**4*K2**2 + 3232*K1**4*K2 - 5552*K1**4 + 2016*K1**3*K2*K3 + 256*K1**3*K3*K4 - 320*K1**3*K3 - 1024*K1**2*K2**4 + 3744*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12064*K1**2*K2**2 - 704*K1**2*K2*K4 + 10024*K1**2*K2 - 1552*K1**2*K3**2 - 32*K1**2*K3*K5 - 384*K1**2*K4**2 - 1772*K1**2 + 2336*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2976*K1*K2**2*K3 - 384*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8720*K1*K2*K3 - 64*K1*K2*K4*K5 + 1624*K1*K3*K4 + 296*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 3080*K2**4 - 32*K2**3*K6 - 1808*K2**2*K3**2 - 320*K2**2*K4**2 + 2448*K2**2*K4 - 2002*K2**2 - 96*K2*K3**2*K4 + 888*K2*K3*K5 + 208*K2*K4*K6 - 1648*K3**2 - 550*K4**2 - 100*K5**2 - 14*K6**2 + 3276
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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