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Flat knot 6.636

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,1,1,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.636']
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+56t^5+37t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.636']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 1376*K1**4 + 96*K1**3*K2*K3 - 672*K1**3*K3 + 32*K1**2*K2**2*K4 - 736*K1**2*K2**2 - 160*K1**2*K2*K4 + 3336*K1**2*K2 - 416*K1**2*K3**2 - 64*K1**2*K4**2 - 2064*K1**2 - 32*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2264*K1*K2*K3 + 448*K1*K3*K4 + 56*K1*K4*K5 - 16*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 184*K2**2*K4 - 1638*K2**2 + 48*K2*K3*K5 + 8*K2*K4*K6 - 744*K3**2 - 160*K4**2 - 16*K5**2 - 2*K6**2 + 1630
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.636']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11006', 'vk6.11087', 'vk6.12174', 'vk6.12283', 'vk6.18203', 'vk6.18540', 'vk6.24661', 'vk6.25087', 'vk6.30569', 'vk6.30666', 'vk6.31841', 'vk6.31890', 'vk6.36791', 'vk6.37245', 'vk6.44032', 'vk6.44374', 'vk6.51817', 'vk6.51886', 'vk6.52683', 'vk6.52779', 'vk6.56009', 'vk6.56284', 'vk6.60548', 'vk6.60890', 'vk6.63495', 'vk6.63541', 'vk6.63975', 'vk6.64021', 'vk6.65668', 'vk6.65954', 'vk6.68714', 'vk6.68924', 'vk6.83170', 'vk6.83588', 'vk6.84138', 'vk6.84331', 'vk6.86469', 'vk6.86483', 'vk6.88739', 'vk6.88906']
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 O1O2O3O4U1O5U3U5O6U2U4U6
R3 orbit {'O1O2O3O4U1O5U3U5O6U2U4U6', 'O1O2O3O4U5U2O6U4U1U3O5U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U5U1U3O5U6U2O6U4
Gauss code of K* O1O2O3U4U1U5U2O4U6O5O6U3
Gauss code of -K* O1O2O3U1O4O5U4O6U2U5U3U6
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 2 1 3 1 2],[ 1 -2 0 -1 2 1 2],[ 1 -1 1 0 2 1 1],[-2 -3 -2 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 0 -2 -2 -3],[-2 -1 0 0 -1 -2 -2],[-1 0 0 0 -1 -1 -1],[ 1 2 1 1 0 1 -1],[ 1 2 2 1 -1 0 -2],[ 3 3 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,0,2,2,3,0,1,2,2,1,1,1,-1,1,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,1,1,1,2,1,1,-1]
Phi of -K [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,1,1,1,2,1,1,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,1,1,2,3,1,1,1,2,1,1,3,-1,0,1]
Phi of -K* [-3,-1,-1,1,2,2,1,2,1,2,3,1,1,1,2,1,2,2,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial z^2+14z+25
Enhanced Jones-Krushkal polynomial w^3z^2+14w^2z+25w
Inner characteristic polynomial t^6+36t^4+17t^2+1
Outer characteristic polynomial t^7+56t^5+37t^3+4t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial 96*K1**4*K2 - 1376*K1**4 + 96*K1**3*K2*K3 - 672*K1**3*K3 + 32*K1**2*K2**2*K4 - 736*K1**2*K2**2 - 160*K1**2*K2*K4 + 3336*K1**2*K2 - 416*K1**2*K3**2 - 64*K1**2*K4**2 - 2064*K1**2 - 32*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2264*K1*K2*K3 + 448*K1*K3*K4 + 56*K1*K4*K5 - 16*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 184*K2**2*K4 - 1638*K2**2 + 48*K2*K3*K5 + 8*K2*K4*K6 - 744*K3**2 - 160*K4**2 - 16*K5**2 - 2*K6**2 + 1630
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
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