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Flat knot 6.635

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,0,2,2,3,1,1,2,2,-1,-1,-1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.635']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+47t^5+57t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.635']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 1344*K1**4*K2 - 2208*K1**4 + 448*K1**3*K2*K3 - 256*K1**3*K3 - 1216*K1**2*K2**4 + 3264*K1**2*K2**3 - 8288*K1**2*K2**2 - 448*K1**2*K2*K4 + 8040*K1**2*K2 - 96*K1**2*K3**2 - 4296*K1**2 + 1376*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5832*K1*K2*K3 + 368*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 2160*K2**4 - 448*K2**2*K3**2 - 8*K2**2*K4**2 + 1328*K2**2*K4 - 2272*K2**2 + 128*K2*K3*K5 - 1244*K3**2 - 276*K4**2 - 4*K5**2 + 3386
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.635']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10913', 'vk6.10922', 'vk6.10936', 'vk6.12077', 'vk6.12087', 'vk6.12092', 'vk6.12102', 'vk6.14482', 'vk6.14493', 'vk6.15703', 'vk6.15716', 'vk6.16143', 'vk6.16150', 'vk6.30511', 'vk6.30531', 'vk6.30539', 'vk6.30561', 'vk6.31795', 'vk6.34069', 'vk6.34162', 'vk6.34180', 'vk6.34505', 'vk6.51752', 'vk6.51765', 'vk6.52629', 'vk6.54138', 'vk6.54143', 'vk6.54327', 'vk6.54530', 'vk6.63468', 'vk6.63477', 'vk6.63483']
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 O1O2O3O4U1O5U3U4O6U5U6U2
R3 orbit {'O1O2O3O4U1O5U3U4O6U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U6O5U1U2O6U4
Gauss code of K* O1O2O3U4U3U5U6O4U1O5O6U2
Gauss code of -K* O1O2O3U2O4O5U3O6U4U5U1U6
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 -1 1 1 1],[ 3 0 3 1 2 2 0],[-1 -3 0 -2 0 1 1],[ 1 -1 2 0 1 2 1],[-1 -2 0 -1 0 1 1],[-1 -2 -1 -2 -1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 0 -1 -2],[-1 -1 0 1 -1 -2 -2],[-1 -1 -1 0 -1 -1 0],[-1 0 1 1 0 -2 -3],[ 1 1 2 1 2 0 -1],[ 3 2 2 0 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,0,1,2,-1,1,2,2,1,1,0,2,3,1]
Phi over symmetry [-3,-1,1,1,1,1,1,0,2,2,3,1,1,2,2,-1,-1,-1,1,0,-1]
Phi of -K [-3,-1,1,1,1,1,1,1,2,2,4,0,0,1,1,-1,0,-1,1,-1,-1]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,-1,1,4,-1,-1,0,2,0,0,1,1,2,1]
Phi of -K* [-3,-1,1,1,1,1,1,0,2,2,3,1,1,2,2,-1,-1,-1,1,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+33t^4+15t^2
Outer characteristic polynomial t^7+47t^5+57t^3+8t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 1344*K1**4*K2 - 2208*K1**4 + 448*K1**3*K2*K3 - 256*K1**3*K3 - 1216*K1**2*K2**4 + 3264*K1**2*K2**3 - 8288*K1**2*K2**2 - 448*K1**2*K2*K4 + 8040*K1**2*K2 - 96*K1**2*K3**2 - 4296*K1**2 + 1376*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5832*K1*K2*K3 + 368*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 2160*K2**4 - 448*K2**2*K3**2 - 8*K2**2*K4**2 + 1328*K2**2*K4 - 2272*K2**2 + 128*K2*K3*K5 - 1244*K3**2 - 276*K4**2 - 4*K5**2 + 3386
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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