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Flat knot 6.591

Min(phi) over symmetries of the knot is: [-3,-2,2,3,0,2,4,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.591', '7.10949']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^5+51t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.591']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 832*K1**4*K2 - 6304*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 5792*K1**2*K2**2 - 800*K1**2*K2*K4 + 14248*K1**2*K2 - 992*K1**2*K3**2 - 64*K1**2*K3*K5 - 224*K1**2*K4**2 - 8140*K1**2 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 9760*K1*K2*K3 + 2408*K1*K3*K4 + 368*K1*K4*K5 + 16*K1*K5*K6 - 368*K2**4 - 96*K2**2*K3**2 - 16*K2**2*K4**2 + 1224*K2**2*K4 - 6964*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 3420*K3**2 - 1144*K4**2 - 160*K5**2 - 12*K6**2 + 7262
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.591']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20026', 'vk6.20065', 'vk6.21298', 'vk6.21345', 'vk6.27077', 'vk6.27130', 'vk6.28782', 'vk6.28817', 'vk6.38466', 'vk6.38523', 'vk6.40655', 'vk6.40718', 'vk6.45350', 'vk6.45423', 'vk6.47119', 'vk6.47163', 'vk6.56825', 'vk6.56886', 'vk6.57959', 'vk6.58022', 'vk6.61343', 'vk6.61416', 'vk6.62519', 'vk6.62571', 'vk6.66537', 'vk6.66586', 'vk6.67326', 'vk6.67375', 'vk6.69183', 'vk6.69238', 'vk6.69934', 'vk6.69977']
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 O1O2O3O4U2O5U1O6U3U6U5U4
R3 orbit {'O1O2O3O4U2O5U1O6U3U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U6U2O6U4O5U3
Gauss code of K* O1O2O3O4U5U6U1U4O6U3O5U2
Gauss code of -K* O1O2O3O4U3O5U2O6U1U4U6U5
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 3 2 1],[ 3 0 0 2 4 2 1],[ 2 0 0 1 2 1 1],[ 1 -2 -1 0 3 2 1],[-3 -4 -2 -3 0 0 0],[-2 -2 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 2 -2 -3],[-3 0 0 -2 -4],[-2 0 0 -1 -2],[ 2 2 1 0 0],[ 3 4 2 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-2,2,3,0,2,4,1,2,0]
Phi over symmetry [-3,-2,2,3,0,2,4,1,2,0]
Phi of -K [-3,-2,2,3,1,3,2,3,3,1]
Phi of K* [-3,-2,2,3,1,3,2,3,3,1]
Phi of -K* [-3,-2,2,3,0,2,4,1,2,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^4+25t^2
Outer characteristic polynomial t^5+51t^3+10t
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -64*K1**6 - 64*K1**4*K2**2 + 832*K1**4*K2 - 6304*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 5792*K1**2*K2**2 - 800*K1**2*K2*K4 + 14248*K1**2*K2 - 992*K1**2*K3**2 - 64*K1**2*K3*K5 - 224*K1**2*K4**2 - 8140*K1**2 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 9760*K1*K2*K3 + 2408*K1*K3*K4 + 368*K1*K4*K5 + 16*K1*K5*K6 - 368*K2**4 - 96*K2**2*K3**2 - 16*K2**2*K4**2 + 1224*K2**2*K4 - 6964*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 3420*K3**2 - 1144*K4**2 - 160*K5**2 - 12*K6**2 + 7262
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice True
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