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Flat knot 6.474

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,1,1,1,1,0,1,1,1,0,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.474', '7.25961']
Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.474', '6.1684', '6.1716', '6.1749', '6.1781']
Outer characteristic polynomial of the knot is: t^7+32t^5+60t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.474']
2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 1984*K1**4*K2**2 + 3488*K1**4*K2 - 6256*K1**4 + 2144*K1**3*K2*K3 + 256*K1**3*K3*K4 - 672*K1**3*K3 - 1024*K1**2*K2**4 + 3744*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12640*K1**2*K2**2 - 1024*K1**2*K2*K4 + 10760*K1**2*K2 - 1904*K1**2*K3**2 - 32*K1**2*K3*K5 - 480*K1**2*K4**2 - 1452*K1**2 + 2336*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2336*K1*K2**2*K3 - 448*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9392*K1*K2*K3 - 96*K1*K2*K4*K5 + 1784*K1*K3*K4 + 328*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 3272*K2**4 - 32*K2**3*K6 - 1776*K2**2*K3**2 - 384*K2**2*K4**2 + 2576*K2**2*K4 - 1906*K2**2 - 64*K2*K3**2*K4 + 984*K2*K3*K5 + 240*K2*K4*K6 - 1648*K3**2 - 550*K4**2 - 100*K5**2 - 14*K6**2 + 3276
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.474']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.509', 'vk6.601', 'vk6.627', 'vk6.1006', 'vk6.1104', 'vk6.1136', 'vk6.1672', 'vk6.1844', 'vk6.2172', 'vk6.2188', 'vk6.2281', 'vk6.2315', 'vk6.2785', 'vk6.2887', 'vk6.3059', 'vk6.3189', 'vk6.5256', 'vk6.6513', 'vk6.8893', 'vk6.9810', 'vk6.20820', 'vk6.21055', 'vk6.22216', 'vk6.22477', 'vk6.28506', 'vk6.29779', 'vk6.39873', 'vk6.40282', 'vk6.46425', 'vk6.46917', 'vk6.49144', 'vk6.58831']
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 O1O2O3O4O5U6U4O6U3U5U1U2
R3 orbit {'O1O2O3O4O5U6U4O6U3U5U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U5U1U3O6U2U6
Gauss code of K* O1O2O3O4U3U4U1U5U2O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U3U6U4U1U2
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 -1 1 -1 0 2 -1],[ 1 0 1 -1 0 2 0],[-1 -1 0 -1 0 2 -2],[ 1 1 1 0 1 2 0],[ 0 0 0 -1 0 0 0],[-2 -2 -2 -2 0 0 -2],[ 1 0 2 0 0 2 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -2 0 -2 -2 -2],[-1 2 0 0 -1 -1 -2],[ 0 0 0 0 0 -1 0],[ 1 2 1 0 0 -1 0],[ 1 2 1 1 1 0 0],[ 1 2 2 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,2,0,2,2,2,0,1,1,2,0,1,0,1,0,0]
Phi over symmetry [-2,-1,0,1,1,1,-1,2,1,1,1,1,0,1,1,1,0,1,0,0,1]
Phi of -K [-1,-1,-1,0,1,2,-1,0,0,1,1,0,1,1,1,1,0,1,1,2,-1]
Phi of K* [-2,-1,0,1,1,1,-1,2,1,1,1,1,0,1,1,1,0,1,0,0,1]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,2,0,2,2,0,0,2]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+24t^4+37t^2+4
Outer characteristic polynomial t^7+32t^5+60t^3+9t
Flat arrow polynomial 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
2-strand cable arrow polynomial -1024*K1**6 - 1984*K1**4*K2**2 + 3488*K1**4*K2 - 6256*K1**4 + 2144*K1**3*K2*K3 + 256*K1**3*K3*K4 - 672*K1**3*K3 - 1024*K1**2*K2**4 + 3744*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12640*K1**2*K2**2 - 1024*K1**2*K2*K4 + 10760*K1**2*K2 - 1904*K1**2*K3**2 - 32*K1**2*K3*K5 - 480*K1**2*K4**2 - 1452*K1**2 + 2336*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2336*K1*K2**2*K3 - 448*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9392*K1*K2*K3 - 96*K1*K2*K4*K5 + 1784*K1*K3*K4 + 328*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 3272*K2**4 - 32*K2**3*K6 - 1776*K2**2*K3**2 - 384*K2**2*K4**2 + 2576*K2**2*K4 - 1906*K2**2 - 64*K2*K3**2*K4 + 984*K2*K3*K5 + 240*K2*K4*K6 - 1648*K3**2 - 550*K4**2 - 100*K5**2 - 14*K6**2 + 3276
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice False
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