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Flat knot 6.612

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,1,1,0,1,-1,1,0,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.612', '7.36864']
Arrow polynomial of the knot is: -12*K1**2 + 6*K2 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.612', '6.1246', '6.1402', '6.1410', '6.1411', '6.1468', '6.1617', '6.1666', '6.1667', '6.1815', '6.1904', '6.1994', '6.1995', '6.1996', '6.2001', '6.2014', '6.2015', '6.2016', '6.2017', '6.2020', '6.2022']
Outer characteristic polynomial of the knot is: t^7+14t^5+23t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.612', '7.36864']
2-strand cable arrow polynomial of the knot is: -1536*K1**6 - 1664*K1**4*K2**2 + 3904*K1**4*K2 - 8224*K1**4 + 1088*K1**3*K2*K3 - 320*K1**3*K3 + 1664*K1**2*K2**3 - 10048*K1**2*K2**2 - 320*K1**2*K2*K4 + 10912*K1**2*K2 - 128*K1**2*K3**2 + 576*K1**2 - 832*K1*K2**2*K3 + 5360*K1*K2*K3 + 48*K1*K3*K4 - 1264*K2**4 + 848*K2**2*K4 - 1712*K2**2 - 480*K3**2 - 52*K4**2 + 2178
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.612']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.23', 'vk6.38', 'vk6.157', 'vk6.172', 'vk6.1206', 'vk6.1299', 'vk6.1310', 'vk6.2358', 'vk6.2391', 'vk6.2955', 'vk6.3525', 'vk6.6909', 'vk6.6940', 'vk6.15389', 'vk6.15505', 'vk6.33450', 'vk6.33507', 'vk6.33612', 'vk6.49938', 'vk6.53756']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U3O5U4O6U5U6U1U2
R3 orbit {'O1O2O3O4U3O5U4O6U5U6U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U4U5U6O5U1O6U2
Gauss code of K* O1O2O3O4U3U4U5U6O5U1O6U2
Gauss code of -K* Same
Diagrammatic symmetry type -
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 1 -1 0 0 1],[ 1 0 1 -1 0 0 1],[-1 -1 0 -1 0 0 1],[ 1 1 1 0 1 1 0],[ 0 0 0 -1 0 1 1],[ 0 0 0 -1 -1 0 1],[-1 -1 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 0 -1 -1],[-1 -1 0 -1 -1 0 -1],[ 0 0 1 0 1 -1 0],[ 0 0 1 -1 0 -1 0],[ 1 1 0 1 1 0 1],[ 1 1 1 0 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,0,1,1,1,1,0,1,-1,1,0,1,0,-1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,1,1,0,1,-1,1,0,1,0,-1]
Phi of -K [-1,-1,0,0,1,1,-1,0,0,1,2,1,1,1,1,-1,1,0,1,0,-1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,1,2,1,1,1,1,-1,1,0,1,0,-1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,0,1,1,1,1,0,1,-1,1,0,1,0,-1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+10t^4+9t^2
Outer characteristic polynomial t^7+14t^5+23t^3+2t
Flat arrow polynomial -12*K1**2 + 6*K2 + 7
2-strand cable arrow polynomial -1536*K1**6 - 1664*K1**4*K2**2 + 3904*K1**4*K2 - 8224*K1**4 + 1088*K1**3*K2*K3 - 320*K1**3*K3 + 1664*K1**2*K2**3 - 10048*K1**2*K2**2 - 320*K1**2*K2*K4 + 10912*K1**2*K2 - 128*K1**2*K3**2 + 576*K1**2 - 832*K1*K2**2*K3 + 5360*K1*K2*K3 + 48*K1*K3*K4 - 1264*K2**4 + 848*K2**2*K4 - 1712*K2**2 - 480*K3**2 - 52*K4**2 + 2178
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}]]
If K is slice True
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