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Flat knot 6.674

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,2,2,4,0,0,1,1,1,2,3,1,0,-2]
Flat knots (up to 7 crossings) with same phi are :['6.674']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^7+66t^5+66t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.674']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 352*K1**4*K2 - 3376*K1**4 + 160*K1**3*K2*K3 - 288*K1**3*K3 + 640*K1**2*K2**3 - 7280*K1**2*K2**2 - 480*K1**2*K2*K4 + 9152*K1**2*K2 - 144*K1**2*K3**2 - 3800*K1**2 + 352*K1*K2**3*K3 - 640*K1*K2**2*K3 - 224*K1*K2**2*K5 + 6768*K1*K2*K3 + 632*K1*K3*K4 + 72*K1*K4*K5 - 2256*K2**4 - 432*K2**2*K3**2 - 8*K2**2*K4**2 + 1928*K2**2*K4 - 2822*K2**2 + 320*K2*K3*K5 + 8*K2*K4*K6 - 1516*K3**2 - 512*K4**2 - 76*K5**2 - 2*K6**2 + 3662
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.674']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17089', 'vk6.17331', 'vk6.20245', 'vk6.21552', 'vk6.23476', 'vk6.23815', 'vk6.27468', 'vk6.29065', 'vk6.35617', 'vk6.36063', 'vk6.38879', 'vk6.41081', 'vk6.42990', 'vk6.43301', 'vk6.45644', 'vk6.47381', 'vk6.55228', 'vk6.55479', 'vk6.57078', 'vk6.58233', 'vk6.59629', 'vk6.59974', 'vk6.61616', 'vk6.62796', 'vk6.65026', 'vk6.65228', 'vk6.66708', 'vk6.67567', 'vk6.68298', 'vk6.68447', 'vk6.69358', 'vk6.70102']
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 O1O2O3O4U3O5U1U2O6U5U4U6
R3 orbit {'O1O2O3O4U3O5U1U2O6U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U6O5U3U4O6U2
Gauss code of K* O1O2O3U4U5U6U2O6U1O4O5U3
Gauss code of -K* O1O2O3U1O4O5U3O6U2U6U4U5
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 -1 -1 2 1 2],[ 3 0 1 0 4 2 2],[ 1 -1 0 0 3 1 2],[ 1 0 0 0 1 0 1],[-2 -4 -3 -1 0 0 2],[-1 -2 -1 0 0 0 1],[-2 -2 -2 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 2 0 -1 -3 -4],[-2 -2 0 -1 -1 -2 -2],[-1 0 1 0 0 -1 -2],[ 1 1 1 0 0 0 0],[ 1 3 2 1 0 0 -1],[ 3 4 2 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-2,0,1,3,4,1,1,2,2,0,1,2,0,0,1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,2,2,4,0,0,1,1,1,2,3,1,0,-2]
Phi of -K [-3,-1,-1,1,2,2,1,2,2,1,3,0,1,0,1,2,2,2,1,0,-2]
Phi of K* [-2,-2,-1,1,1,3,-2,0,1,2,3,1,0,2,1,1,2,2,0,1,2]
Phi of -K* [-3,-1,-1,1,2,2,0,1,2,2,4,0,0,1,1,1,2,3,1,0,-2]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+46t^4+24t^2+1
Outer characteristic polynomial t^7+66t^5+66t^3+10t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -192*K1**4*K2**2 + 352*K1**4*K2 - 3376*K1**4 + 160*K1**3*K2*K3 - 288*K1**3*K3 + 640*K1**2*K2**3 - 7280*K1**2*K2**2 - 480*K1**2*K2*K4 + 9152*K1**2*K2 - 144*K1**2*K3**2 - 3800*K1**2 + 352*K1*K2**3*K3 - 640*K1*K2**2*K3 - 224*K1*K2**2*K5 + 6768*K1*K2*K3 + 632*K1*K3*K4 + 72*K1*K4*K5 - 2256*K2**4 - 432*K2**2*K3**2 - 8*K2**2*K4**2 + 1928*K2**2*K4 - 2822*K2**2 + 320*K2*K3*K5 + 8*K2*K4*K6 - 1516*K3**2 - 512*K4**2 - 76*K5**2 - 2*K6**2 + 3662
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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