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Flat knot 6.446

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,1,3,1,1,1,2,1,1,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.446']
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+43t^5+41t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.446']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 832*K1**4*K2**2 + 2944*K1**4*K2 - 5584*K1**4 + 1344*K1**3*K2*K3 - 1568*K1**3*K3 + 2720*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11120*K1**2*K2**2 - 1216*K1**2*K2*K4 + 11880*K1**2*K2 - 1264*K1**2*K3**2 - 320*K1**2*K4**2 - 4064*K1**2 + 1344*K1*K2**3*K3 - 2080*K1*K2**2*K3 - 288*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 9776*K1*K2*K3 + 1608*K1*K3*K4 + 264*K1*K4*K5 - 32*K2**6 + 416*K2**4*K4 - 2952*K2**4 - 64*K2**3*K6 - 1616*K2**2*K3**2 - 544*K2**2*K4**2 + 2488*K2**2*K4 - 3234*K2**2 + 968*K2*K3*K5 + 232*K2*K4*K6 - 1996*K3**2 - 622*K4**2 - 132*K5**2 - 14*K6**2 + 4324
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.446']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4693', 'vk6.4998', 'vk6.6179', 'vk6.6652', 'vk6.8172', 'vk6.8592', 'vk6.9558', 'vk6.9899', 'vk6.17391', 'vk6.20909', 'vk6.20986', 'vk6.22321', 'vk6.22408', 'vk6.23560', 'vk6.23899', 'vk6.28389', 'vk6.36159', 'vk6.40039', 'vk6.40187', 'vk6.42092', 'vk6.43072', 'vk6.43378', 'vk6.46571', 'vk6.46696', 'vk6.48725', 'vk6.49513', 'vk6.49718', 'vk6.51423', 'vk6.55557', 'vk6.58909', 'vk6.65295', 'vk6.69765']
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 O1O2O3O4O5U6U2O6U4U5U1U3
R3 orbit {'O1O2O3O4O5U6U2O6U4U5U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U5U1U2O6U4U6
Gauss code of K* O1O2O3O4U3U5U4U1U2O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U3U4U1U6U2
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 -1 -2 2 0 2 -1],[ 1 0 -1 2 0 2 0],[ 2 1 0 2 0 1 2],[-2 -2 -2 0 -1 1 -2],[ 0 0 0 1 0 1 0],[-2 -2 -1 -1 -1 0 -2],[ 1 0 -2 2 0 2 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 1 -1 -2 -2 -2],[-2 -1 0 -1 -2 -2 -1],[ 0 1 1 0 0 0 0],[ 1 2 2 0 0 0 -1],[ 1 2 2 0 0 0 -2],[ 2 2 1 0 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,-1,1,2,2,2,1,2,2,1,0,0,0,0,1,2]
Phi over symmetry [-2,-2,0,1,1,2,-1,1,1,1,3,1,1,1,2,1,1,2,0,-1,0]
Phi of -K [-2,-1,-1,0,2,2,-1,0,2,2,3,0,1,1,1,1,1,1,1,1,-1]
Phi of K* [-2,-2,0,1,1,2,-1,1,1,1,3,1,1,1,2,1,1,2,0,-1,0]
Phi of -K* [-2,-1,-1,0,2,2,1,2,0,1,2,0,0,2,2,0,2,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+29t^4+20t^2
Outer characteristic polynomial t^7+43t^5+41t^3+5t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -512*K1**6 - 832*K1**4*K2**2 + 2944*K1**4*K2 - 5584*K1**4 + 1344*K1**3*K2*K3 - 1568*K1**3*K3 + 2720*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11120*K1**2*K2**2 - 1216*K1**2*K2*K4 + 11880*K1**2*K2 - 1264*K1**2*K3**2 - 320*K1**2*K4**2 - 4064*K1**2 + 1344*K1*K2**3*K3 - 2080*K1*K2**2*K3 - 288*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 9776*K1*K2*K3 + 1608*K1*K3*K4 + 264*K1*K4*K5 - 32*K2**6 + 416*K2**4*K4 - 2952*K2**4 - 64*K2**3*K6 - 1616*K2**2*K3**2 - 544*K2**2*K4**2 + 2488*K2**2*K4 - 3234*K2**2 + 968*K2*K3*K5 + 232*K2*K4*K6 - 1996*K3**2 - 622*K4**2 - 132*K5**2 - 14*K6**2 + 4324
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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