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Flat knot 6.1595

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,1,0,1,0,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1595']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+30t^5+71t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1595']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 2272*K1**4*K2 - 4608*K1**4 - 256*K1**3*K2**2*K3 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1120*K1**3*K3 + 1376*K1**2*K2**3 - 6768*K1**2*K2**2 + 288*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 9896*K1**2*K2 - 1504*K1**2*K3**2 - 224*K1**2*K3*K5 - 144*K1**2*K4**2 - 5808*K1**2 + 480*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 8040*K1*K2*K3 - 32*K1*K3**2*K5 + 2520*K1*K3*K4 + 400*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 968*K2**4 - 624*K2**2*K3**2 - 64*K2**2*K4**2 + 1184*K2**2*K4 - 4482*K2**2 + 560*K2*K3*K5 + 56*K2*K4*K6 - 32*K3**4 + 40*K3**2*K6 - 2656*K3**2 - 1078*K4**2 - 248*K5**2 - 30*K6**2 + 5348
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1595']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4822', 'vk6.5165', 'vk6.6386', 'vk6.6817', 'vk6.8351', 'vk6.8783', 'vk6.9723', 'vk6.10026', 'vk6.11630', 'vk6.11983', 'vk6.12976', 'vk6.20472', 'vk6.20723', 'vk6.21827', 'vk6.27856', 'vk6.29366', 'vk6.31433', 'vk6.32611', 'vk6.39294', 'vk6.39763', 'vk6.41474', 'vk6.46327', 'vk6.47593', 'vk6.47904', 'vk6.49047', 'vk6.49875', 'vk6.51309', 'vk6.51526', 'vk6.53241', 'vk6.57331', 'vk6.62017', 'vk6.64318']
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 O1O2O3U4O5U6U3O6O4U1U5U2
R3 orbit {'O1O2O3U4O5U6U3O6O4U1U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U3O5O6U1U6O4U5
Gauss code of K* O1O2O3U1U3U4O5U2O6O4U6U5
Gauss code of -K* O1O2O3U4U5O6O5U2O4U6U1U3
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 -2 1 1 0 1 -1],[ 2 0 2 2 1 1 1],[-1 -2 0 1 -1 0 -2],[-1 -2 -1 0 0 -1 -1],[ 0 -1 1 0 0 1 -1],[-1 -1 0 1 -1 0 -1],[ 1 -1 2 1 1 1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 0 -1 -2],[-1 0 1 0 -1 -2 -2],[ 0 1 0 1 0 -1 -1],[ 1 1 1 2 1 0 -1],[ 2 1 2 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,1,1,1,1,0,1,2,1,2,2,1,1,1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,1,0,1,0,-1,0,1]
Phi of -K [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,1,0,1,0,-1,0,1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,1,1,1,0,0,0,1,0,1,2,0,1,0]
Phi of -K* [-2,-1,0,1,1,1,1,1,1,2,2,1,1,1,2,1,0,1,1,0,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+21z+35
Enhanced Jones-Krushkal polynomial 2w^3z^2-2w^3z+23w^2z+35w
Inner characteristic polynomial t^6+22t^4+46t^2+1
Outer characteristic polynomial t^7+30t^5+71t^3+10t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -128*K1**6 - 256*K1**4*K2**2 + 2272*K1**4*K2 - 4608*K1**4 - 256*K1**3*K2**2*K3 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1120*K1**3*K3 + 1376*K1**2*K2**3 - 6768*K1**2*K2**2 + 288*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 9896*K1**2*K2 - 1504*K1**2*K3**2 - 224*K1**2*K3*K5 - 144*K1**2*K4**2 - 5808*K1**2 + 480*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 8040*K1*K2*K3 - 32*K1*K3**2*K5 + 2520*K1*K3*K4 + 400*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 968*K2**4 - 624*K2**2*K3**2 - 64*K2**2*K4**2 + 1184*K2**2*K4 - 4482*K2**2 + 560*K2*K3*K5 + 56*K2*K4*K6 - 32*K3**4 + 40*K3**2*K6 - 2656*K3**2 - 1078*K4**2 - 248*K5**2 - 30*K6**2 + 5348
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice False
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