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Flat knot 6.527

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,1,3,4,3,1,2,2,1,1,1,2,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.527']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+62t^5+65t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.527']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 256*K1**4*K2**2 + 992*K1**4*K2 - 2176*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 832*K1**3*K3 - 320*K1**2*K2**4 + 768*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 3584*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 6968*K1**2*K2 - 576*K1**2*K3**2 - 128*K1**2*K4**2 - 4836*K1**2 + 480*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 128*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 5272*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1448*K1*K3*K4 + 320*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 552*K2**4 - 352*K2**2*K3**2 - 96*K2**2*K4**2 + 1080*K2**2*K4 - 3618*K2**2 + 464*K2*K3*K5 + 56*K2*K4*K6 + 40*K3**2*K6 - 1944*K3**2 - 782*K4**2 - 236*K5**2 - 46*K6**2 + 3956
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.527']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4217', 'vk6.4297', 'vk6.5481', 'vk6.5594', 'vk6.7584', 'vk6.7679', 'vk6.9089', 'vk6.9169', 'vk6.11176', 'vk6.12258', 'vk6.12367', 'vk6.19365', 'vk6.19658', 'vk6.19784', 'vk6.26149', 'vk6.26219', 'vk6.26565', 'vk6.26662', 'vk6.30766', 'vk6.31967', 'vk6.38149', 'vk6.38207', 'vk6.44810', 'vk6.44948', 'vk6.48527', 'vk6.49225', 'vk6.49336', 'vk6.50316', 'vk6.52744', 'vk6.63582', 'vk6.66317', 'vk6.66347']
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 O1O2O3O4O5U2U5U3O6U1U6U4
R3 orbit {'O1O2O3O4O5U2U5U3O6U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U5O6U3U1U4
Gauss code of K* O1O2O3U4O5O4O6U5U1U3U6U2
Gauss code of -K* O1O2O3U2O4O5O6U5U1U4U6U3
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 -3 0 3 1 1],[ 2 0 -2 1 4 1 1],[ 3 2 0 2 3 1 0],[ 0 -1 -2 0 1 0 0],[-3 -4 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -2 -3],[-3 0 0 0 -1 -4 -3],[-1 0 0 0 0 -1 0],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -2],[ 2 4 1 1 1 0 -2],[ 3 3 0 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,2,3,0,0,1,4,3,0,0,1,0,0,1,1,1,2,2]
Phi over symmetry [-3,-2,0,1,1,3,-1,1,3,4,3,1,2,2,1,1,1,2,0,2,2]
Phi of -K [-3,-2,0,1,1,3,-1,1,3,4,3,1,2,2,1,1,1,2,0,2,2]
Phi of K* [-3,-1,-1,0,2,3,2,2,2,1,3,0,1,2,3,1,2,4,1,1,-1]
Phi of -K* [-3,-2,0,1,1,3,2,2,0,1,3,1,1,1,4,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+38t^4+28t^2+1
Outer characteristic polynomial t^7+62t^5+65t^3+5t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -192*K1**6 - 256*K1**4*K2**2 + 992*K1**4*K2 - 2176*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 832*K1**3*K3 - 320*K1**2*K2**4 + 768*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 3584*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 6968*K1**2*K2 - 576*K1**2*K3**2 - 128*K1**2*K4**2 - 4836*K1**2 + 480*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 128*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 5272*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1448*K1*K3*K4 + 320*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 552*K2**4 - 352*K2**2*K3**2 - 96*K2**2*K4**2 + 1080*K2**2*K4 - 3618*K2**2 + 464*K2*K3*K5 + 56*K2*K4*K6 + 40*K3**2*K6 - 1944*K3**2 - 782*K4**2 - 236*K5**2 - 46*K6**2 + 3956
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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