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Flat knot 6.740

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,4,2,1,1,2,1,0,-1,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.740']
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+89t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.740']
2-strand cable arrow polynomial of the knot is: -736*K1**4 + 32*K1**3*K2*K3 - 160*K1**3*K3 + 96*K1**2*K2**3 - 2016*K1**2*K2**2 - 128*K1**2*K2*K4 + 4744*K1**2*K2 - 320*K1**2*K3**2 - 3968*K1**2 - 576*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3504*K1*K2*K3 + 824*K1*K3*K4 + 64*K1*K4*K5 - 80*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 440*K2**2*K4 - 2854*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 1368*K3**2 - 400*K4**2 - 112*K5**2 - 18*K6**2 + 2902
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.740']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11231', 'vk6.11312', 'vk6.12496', 'vk6.12609', 'vk6.18211', 'vk6.18547', 'vk6.24675', 'vk6.25097', 'vk6.30909', 'vk6.31034', 'vk6.32097', 'vk6.32218', 'vk6.36805', 'vk6.37263', 'vk6.44049', 'vk6.44390', 'vk6.51993', 'vk6.52090', 'vk6.52874', 'vk6.52923', 'vk6.56016', 'vk6.56291', 'vk6.60561', 'vk6.60902', 'vk6.63648', 'vk6.63695', 'vk6.64080', 'vk6.64127', 'vk6.65681', 'vk6.65972', 'vk6.68730', 'vk6.68939']
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 O1O2O3O4U2O5U1U4U5O6U3U6
R3 orbit {'O1O2O3O4U2O5U1U4U5O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2O5U6U1U4O6U3
Gauss code of K* O1O2O3U4O5O4U1U6U5U2O6U3
Gauss code of -K* O1O2O3U1O4U2U5U4U3O6O5U6
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 -2 1 1 2 1],[ 3 0 0 4 2 2 1],[ 2 0 0 2 1 1 1],[-1 -4 -2 0 -1 1 1],[-1 -2 -1 1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 0 -1 -1 -1 -2],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 1 -1 -2],[-1 1 1 -1 0 -2 -4],[ 2 1 1 1 2 0 0],[ 3 2 1 2 4 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,0,1,1,1,2,0,1,1,1,-1,1,2,2,4,0]
Phi over symmetry [-3,-2,1,1,1,2,0,1,2,4,2,1,1,2,1,0,-1,0,1,1,1]
Phi of -K [-3,-2,1,1,1,2,1,0,2,3,3,1,2,2,3,1,-1,0,0,0,1]
Phi of K* [-2,-1,-1,-1,2,3,0,0,1,3,3,-1,1,1,0,0,2,2,2,3,1]
Phi of -K* [-3,-2,1,1,1,2,0,1,2,4,2,1,1,2,1,0,-1,0,1,1,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+36t^4+23t^2
Outer characteristic polynomial t^7+56t^5+89t^3+4t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -736*K1**4 + 32*K1**3*K2*K3 - 160*K1**3*K3 + 96*K1**2*K2**3 - 2016*K1**2*K2**2 - 128*K1**2*K2*K4 + 4744*K1**2*K2 - 320*K1**2*K3**2 - 3968*K1**2 - 576*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3504*K1*K2*K3 + 824*K1*K3*K4 + 64*K1*K4*K5 - 80*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 440*K2**2*K4 - 2854*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 1368*K3**2 - 400*K4**2 - 112*K5**2 - 18*K6**2 + 2902
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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