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Flat knot 6.136

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,1,1,1,2,2,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.136']
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+24t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.136']
2-strand cable arrow polynomial of the knot is: -96*K1**4 - 128*K1**2*K2**2 + 400*K1**2*K2 - 32*K1**2*K3**2 - 344*K1**2 + 208*K1*K2*K3 + 96*K1*K3*K4 - 16*K2**4 - 8*K2**2*K4**2 + 40*K2**2*K4 - 246*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 120*K3**2 - 56*K4**2 - 16*K5**2 - 2*K6**2 + 278
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.136']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4422', 'vk6.4517', 'vk6.5804', 'vk6.5931', 'vk6.6420', 'vk6.6850', 'vk6.7972', 'vk6.8381', 'vk6.9283', 'vk6.9402', 'vk6.17898', 'vk6.17963', 'vk6.18633', 'vk6.24401', 'vk6.24701', 'vk6.25183', 'vk6.30034', 'vk6.30095', 'vk6.30904', 'vk6.31027', 'vk6.32088', 'vk6.32207', 'vk6.36908', 'vk6.37285', 'vk6.37366', 'vk6.39842', 'vk6.43828', 'vk6.44125', 'vk6.44448', 'vk6.47977', 'vk6.49909', 'vk6.50593', 'vk6.51972', 'vk6.52067', 'vk6.60571', 'vk6.65978', 'vk6.72341', 'vk6.77233', 'vk6.78717', 'vk6.78919', 'vk6.84465', 'vk6.87979', 'vk6.88353', 'vk6.89325']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U4U6U3U1U5U2
R3 orbit {'O1O2O3O4U2O5U6U4U1O6U3U5', 'O1O2O3U1O4O5U3U6U4U2O6U5', 'O1O2O3O4O5U3O6U4U6U1U5U2', 'O1O2O3U1O4O5U6U2U4O6U3U5', 'O1O2O3O4O5U3U6U4U1O6U5U2', 'O1O2O3O4O5U3U5U6U1U4O6U2', 'O1O2O3O4O5O6U4U6U3U1U5U2', 'O1O2O3O4U2O5O6U4U6U1U3U5'}
R3 orbit length 8
Gauss code of -K O1O2O3O4O5O6U5U2U6U4U1U3
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U5U2U6U4U1U3
Diagrammatic symmetry type +
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 1 -1 -2 3 1],[ 2 0 2 0 -1 3 1],[-1 -2 0 -1 -2 2 1],[ 1 0 1 0 -1 2 1],[ 2 1 2 1 0 2 1],[-3 -3 -2 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -2 -2 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -1 -2 -2],[ 1 2 1 1 0 -1 0],[ 2 2 1 2 1 0 1],[ 2 3 1 2 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,1,1,1,2,2,1,0,-1]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,1,1,1,2,2,1,0,-1]
Phi of -K [-2,-2,-1,1,1,3,-1,0,1,2,3,1,1,2,2,1,1,2,-1,0,2]
Phi of K* [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,1,1,1,2,2,1,0,-1]
Phi of -K* [-2,-2,-1,1,1,3,-1,0,1,2,3,1,1,2,2,1,1,2,-1,0,2]
Symmetry type of based matrix +
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial 5w^2z+11w
Inner characteristic polynomial t^6+36t^4+8t^2
Outer characteristic polynomial t^7+56t^5+24t^3
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -96*K1**4 - 128*K1**2*K2**2 + 400*K1**2*K2 - 32*K1**2*K3**2 - 344*K1**2 + 208*K1*K2*K3 + 96*K1*K3*K4 - 16*K2**4 - 8*K2**2*K4**2 + 40*K2**2*K4 - 246*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 120*K3**2 - 56*K4**2 - 16*K5**2 - 2*K6**2 + 278
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}]]
If K is slice False
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