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Flat knot 6.339

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,0,2,3,2,4,1,1,1,1,0,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.339']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K1*K3 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.234', '6.339']
Outer characteristic polynomial of the knot is: t^7+77t^5+47t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.339']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 1024*K1**4*K2 - 5040*K1**4 + 768*K1**3*K2*K3 - 864*K1**3*K3 - 256*K1**2*K2**4 + 1184*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9888*K1**2*K2**2 - 1664*K1**2*K2*K4 + 14072*K1**2*K2 - 624*K1**2*K3**2 - 224*K1**2*K4**2 - 7624*K1**2 + 928*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 768*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 11432*K1*K2*K3 - 64*K1*K2*K4*K5 + 1960*K1*K3*K4 + 472*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 2416*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 752*K2**2*K3**2 - 440*K2**2*K4**2 + 3328*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 6456*K2**2 + 808*K2*K3*K5 + 240*K2*K4*K6 + 16*K2*K5*K7 - 3108*K3**2 - 1366*K4**2 - 292*K5**2 - 48*K6**2 + 6972
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.339']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16494', 'vk6.16585', 'vk6.18104', 'vk6.18442', 'vk6.22921', 'vk6.23016', 'vk6.24555', 'vk6.24974', 'vk6.34904', 'vk6.35009', 'vk6.36686', 'vk6.37110', 'vk6.42467', 'vk6.42578', 'vk6.43966', 'vk6.44283', 'vk6.54737', 'vk6.54832', 'vk6.55916', 'vk6.56206', 'vk6.59197', 'vk6.59260', 'vk6.60445', 'vk6.60805', 'vk6.64755', 'vk6.64814', 'vk6.65554', 'vk6.65866', 'vk6.68049', 'vk6.68112', 'vk6.68636', 'vk6.68851']
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:
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invariant value
Gauss code O1O2O3O4O5U3U1O6U4U6U2U5
R3 orbit {'O1O2O3O4O5U3U1O6U4U6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U4U6U2O6U5U3
Gauss code of K* O1O2O3O4U5U3U6U1U4O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U1U4U6U2U5
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 0 -2 0 4 1],[ 3 0 2 0 2 4 1],[ 0 -2 0 -1 0 3 1],[ 2 0 1 0 1 2 1],[ 0 -2 0 -1 0 2 1],[-4 -4 -3 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 4 1 0 0 -2 -3],[-4 0 0 -2 -3 -2 -4],[-1 0 0 -1 -1 -1 -1],[ 0 2 1 0 0 -1 -2],[ 0 3 1 0 0 -1 -2],[ 2 2 1 1 1 0 0],[ 3 4 1 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,0,2,3,0,2,3,2,4,1,1,1,1,0,1,2,1,2,0]
Phi over symmetry [-4,-1,0,0,2,3,0,2,3,2,4,1,1,1,1,0,1,2,1,2,0]
Phi of -K [-3,-2,0,0,1,4,1,1,1,3,3,1,1,2,4,0,0,1,0,2,3]
Phi of K* [-4,-1,0,0,2,3,3,1,2,4,3,0,0,2,3,0,1,1,1,1,1]
Phi of -K* [-3,-2,0,0,1,4,0,2,2,1,4,1,1,1,2,0,1,2,1,3,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+47t^4+18t^2+1
Outer characteristic polynomial t^7+77t^5+47t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K1*K3 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -192*K1**4*K2**2 + 1024*K1**4*K2 - 5040*K1**4 + 768*K1**3*K2*K3 - 864*K1**3*K3 - 256*K1**2*K2**4 + 1184*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9888*K1**2*K2**2 - 1664*K1**2*K2*K4 + 14072*K1**2*K2 - 624*K1**2*K3**2 - 224*K1**2*K4**2 - 7624*K1**2 + 928*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 768*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 11432*K1*K2*K3 - 64*K1*K2*K4*K5 + 1960*K1*K3*K4 + 472*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 2416*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 752*K2**2*K3**2 - 440*K2**2*K4**2 + 3328*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 6456*K2**2 + 808*K2*K3*K5 + 240*K2*K4*K6 + 16*K2*K5*K7 - 3108*K3**2 - 1366*K4**2 - 292*K5**2 - 48*K6**2 + 6972
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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