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Flat knot 6.1837

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,2,1,1,0,1,0,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1837']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 8*K1*K2 + K1 + 3*K2 + 3*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1080', '6.1837', '6.1841', '6.1865']
Outer characteristic polynomial of the knot is: t^7+22t^5+38t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1837']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 944*K1**4 + 384*K1**3*K2*K3 - 352*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 - 5856*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 7592*K1**2*K2 - 304*K1**2*K3**2 - 80*K1**2*K4**2 - 32*K1**2*K5**2 - 6092*K1**2 + 448*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 544*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7904*K1*K2*K3 - 32*K1*K2*K4*K5 + 1272*K1*K3*K4 + 408*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 856*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 64*K2**2*K4**2 + 1704*K2**2*K4 - 4682*K2**2 + 680*K2*K3*K5 + 56*K2*K4*K6 + 8*K3**2*K6 - 2464*K3**2 - 838*K4**2 - 292*K5**2 - 14*K6**2 + 4692
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1837']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11442', 'vk6.11738', 'vk6.12755', 'vk6.13099', 'vk6.20324', 'vk6.21667', 'vk6.27628', 'vk6.29174', 'vk6.31190', 'vk6.31529', 'vk6.32358', 'vk6.32773', 'vk6.39048', 'vk6.41310', 'vk6.45804', 'vk6.47481', 'vk6.52207', 'vk6.52469', 'vk6.53037', 'vk6.53359', 'vk6.57183', 'vk6.58396', 'vk6.61797', 'vk6.62920', 'vk6.63771', 'vk6.63882', 'vk6.64198', 'vk6.64385', 'vk6.66792', 'vk6.67662', 'vk6.69432', 'vk6.70156']
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 O1O2O3U1U4O5U2O6O4U6U3U5
R3 orbit {'O1O2O3U1U4O5U2O6O4U6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1U5O6O5U2O4U6U3
Gauss code of K* O1O2O3U4U5U2O4O6U3O5U1U6
Gauss code of -K* O1O2O3U4U3O5U1O4O6U2U5U6
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 1 2 1 2 0],[ 0 -1 0 0 0 1 -1],[-1 -2 0 0 0 0 -1],[-1 -1 0 0 0 0 -1],[-1 -2 -1 0 0 0 0],[ 1 0 1 1 1 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[-1 0 0 0 -1 0 -2],[ 0 0 0 1 0 -1 -1],[ 1 1 1 0 1 0 0],[ 2 1 2 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,0,0,0,1,1,0,0,1,2,1,0,2,1,1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,2,2,1,1,0,1,0,1,0,0,0,0]
Phi of -K [-2,-1,0,1,1,1,1,1,1,1,2,0,1,2,1,1,0,1,0,0,0]
Phi of K* [-1,-1,-1,0,1,2,0,0,0,2,1,0,1,1,1,1,1,2,0,1,1]
Phi of -K* [-2,-1,0,1,1,1,0,1,1,2,2,1,1,0,1,0,1,0,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+14t^4+21t^2+1
Outer characteristic polynomial t^7+22t^5+38t^3+6t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 8*K1*K2 + K1 + 3*K2 + 3*K3 + 4
2-strand cable arrow polynomial -256*K1**4*K2**2 + 480*K1**4*K2 - 944*K1**4 + 384*K1**3*K2*K3 - 352*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 - 5856*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 7592*K1**2*K2 - 304*K1**2*K3**2 - 80*K1**2*K4**2 - 32*K1**2*K5**2 - 6092*K1**2 + 448*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 544*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7904*K1*K2*K3 - 32*K1*K2*K4*K5 + 1272*K1*K3*K4 + 408*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 856*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 64*K2**2*K4**2 + 1704*K2**2*K4 - 4682*K2**2 + 680*K2*K3*K5 + 56*K2*K4*K6 + 8*K3**2*K6 - 2464*K3**2 - 838*K4**2 - 292*K5**2 - 14*K6**2 + 4692
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}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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