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Flat knot 6.1525

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,3,1,1,1,1,0,1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1525']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1030', '6.1062', '6.1226', '6.1508', '6.1525', '6.1596', '6.1724', '6.1729', '6.1735', '6.1738', '6.1789', '6.1809', '6.1921']
Outer characteristic polynomial of the knot is: t^7+24t^5+39t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1525']
2-strand cable arrow polynomial of the knot is: 1920*K1**4*K2 - 3984*K1**4 + 672*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1920*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4480*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 1344*K1**2*K2*K4 + 8096*K1**2*K2 - 848*K1**2*K3**2 - 128*K1**2*K3*K5 - 144*K1**2*K4**2 - 32*K1**2*K5**2 - 4524*K1**2 + 288*K1*K2**3*K3 - 704*K1*K2**2*K3 - 256*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6720*K1*K2*K3 - 32*K1*K3**2*K5 + 1984*K1*K3*K4 + 488*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 648*K2**4 - 336*K2**2*K3**2 - 64*K2**2*K4**2 + 1480*K2**2*K4 - 3914*K2**2 + 688*K2*K3*K5 + 56*K2*K4*K6 + 8*K3**2*K6 - 2148*K3**2 - 1058*K4**2 - 344*K5**2 - 22*K6**2 + 4136
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1525']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4370', 'vk6.4401', 'vk6.5688', 'vk6.5719', 'vk6.7761', 'vk6.7792', 'vk6.9239', 'vk6.9270', 'vk6.10476', 'vk6.10557', 'vk6.10652', 'vk6.10699', 'vk6.10730', 'vk6.10843', 'vk6.14611', 'vk6.15323', 'vk6.15448', 'vk6.16230', 'vk6.17972', 'vk6.24416', 'vk6.30155', 'vk6.30236', 'vk6.30331', 'vk6.30462', 'vk6.33965', 'vk6.34370', 'vk6.34424', 'vk6.43851', 'vk6.50447', 'vk6.50478', 'vk6.54197', 'vk6.63431']
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 O1O2O3U2O4U5U3O5O6U4U1U6
R3 orbit {'O1O2O3U2O4U5U3O5O6U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3U5O4O6U1U6O5U2
Gauss code of K* O1O2O3U2U4U5O4U1O6O5U6U3
Gauss code of -K* O1O2O3U1U4O5O4U3O6U5U6U2
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 -1 2],[ 1 0 -1 1 1 0 2],[ 1 1 0 1 1 0 0],[-1 -1 -1 0 0 -1 1],[ 0 -1 -1 0 0 0 1],[ 1 0 0 1 0 0 2],[-2 -2 0 -1 -1 -2 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 0 -2 -2],[-1 1 0 0 -1 -1 -1],[ 0 1 0 0 -1 0 -1],[ 1 0 1 1 0 0 1],[ 1 2 1 0 0 0 0],[ 1 2 1 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,1,1,0,2,2,0,1,1,1,1,0,1,0,-1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,3,1,1,1,1,0,1,0,0,-1,0]
Phi of -K [-1,-1,-1,0,1,2,-1,0,0,1,3,0,0,1,1,1,1,1,1,1,0]
Phi of K* [-2,-1,0,1,1,1,0,1,1,1,3,1,1,1,1,0,1,0,0,-1,0]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,1,0,0,1,2,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial 6w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+16t^4+22t^2+1
Outer characteristic polynomial t^7+24t^5+39t^3+6t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 1920*K1**4*K2 - 3984*K1**4 + 672*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1920*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4480*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 1344*K1**2*K2*K4 + 8096*K1**2*K2 - 848*K1**2*K3**2 - 128*K1**2*K3*K5 - 144*K1**2*K4**2 - 32*K1**2*K5**2 - 4524*K1**2 + 288*K1*K2**3*K3 - 704*K1*K2**2*K3 - 256*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6720*K1*K2*K3 - 32*K1*K3**2*K5 + 1984*K1*K3*K4 + 488*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 648*K2**4 - 336*K2**2*K3**2 - 64*K2**2*K4**2 + 1480*K2**2*K4 - 3914*K2**2 + 688*K2*K3*K5 + 56*K2*K4*K6 + 8*K3**2*K6 - 2148*K3**2 - 1058*K4**2 - 344*K5**2 - 22*K6**2 + 4136
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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