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Flat knot 6.164

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,2,4,3,1,1,2,2,1,1,3,0,1,3]
Flat knots (up to 7 crossings) with same phi are :['6.164']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.152', '6.164', '6.256', '6.433']
Outer characteristic polynomial of the knot is: t^7+98t^5+63t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.164']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 1024*K1**4*K2 - 2576*K1**4 + 160*K1**3*K2*K3 - 256*K1**3*K3 + 128*K1**2*K2**5 - 2368*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3808*K1**2*K2**3 - 7904*K1**2*K2**2 - 256*K1**2*K2*K4 + 6984*K1**2*K2 - 144*K1**2*K3**2 - 2736*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2336*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 - 192*K1*K2**2*K5 + 4720*K1*K2*K3 + 232*K1*K3*K4 + 16*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 928*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 832*K2**4*K4 - 2592*K2**4 + 64*K2**3*K3*K5 - 576*K2**2*K3**2 - 136*K2**2*K4**2 + 1296*K2**2*K4 - 736*K2**2 + 136*K2*K3*K5 - 820*K3**2 - 172*K4**2 - 20*K5**2 + 2466
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.164']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17019', 'vk6.17260', 'vk6.20237', 'vk6.21537', 'vk6.23435', 'vk6.23736', 'vk6.27452', 'vk6.29055', 'vk6.35509', 'vk6.35956', 'vk6.38866', 'vk6.41058', 'vk6.42929', 'vk6.43224', 'vk6.45621', 'vk6.47371', 'vk6.55200', 'vk6.55435', 'vk6.57067', 'vk6.58200', 'vk6.59587', 'vk6.59909', 'vk6.61594', 'vk6.62773', 'vk6.64999', 'vk6.65203', 'vk6.66691', 'vk6.67535', 'vk6.68279', 'vk6.68430', 'vk6.69337', 'vk6.70089']
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 O1O2O3O4O5U1O6U4U5U3U6U2
R3 orbit {'O1O2O3O4O5U1O6U4U5U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U3U1U2O6U5
Gauss code of K* O1O2O3O4O5U6U5U3U1U2O6U4
Gauss code of -K* O1O2O3O4O5U2O6U4U5U3U1U6
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 -4 1 0 -1 1 3],[ 4 0 4 3 1 2 3],[-1 -4 0 -1 -2 0 3],[ 0 -3 1 0 -1 1 3],[ 1 -1 2 1 0 1 2],[-1 -2 0 -1 -1 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 1 1 0 -1 -4],[-3 0 -1 -3 -3 -2 -3],[-1 1 0 0 -1 -1 -2],[-1 3 0 0 -1 -2 -4],[ 0 3 1 1 0 -1 -3],[ 1 2 1 2 1 0 -1],[ 4 3 2 4 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,1,4,1,3,3,2,3,0,1,1,2,1,2,4,1,3,1]
Phi over symmetry [-4,-1,0,1,1,3,1,3,2,4,3,1,1,2,2,1,1,3,0,1,3]
Phi of -K [-4,-1,0,1,1,3,2,1,1,3,4,0,0,1,2,0,0,0,0,-1,1]
Phi of K* [-3,-1,-1,0,1,4,-1,1,0,2,4,0,0,0,1,0,1,3,0,1,2]
Phi of -K* [-4,-1,0,1,1,3,1,3,2,4,3,1,1,2,2,1,1,3,0,1,3]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w
Inner characteristic polynomial t^6+70t^4+17t^2
Outer characteristic polynomial t^7+98t^5+63t^3+7t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 1024*K1**4*K2 - 2576*K1**4 + 160*K1**3*K2*K3 - 256*K1**3*K3 + 128*K1**2*K2**5 - 2368*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3808*K1**2*K2**3 - 7904*K1**2*K2**2 - 256*K1**2*K2*K4 + 6984*K1**2*K2 - 144*K1**2*K3**2 - 2736*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2336*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 - 192*K1*K2**2*K5 + 4720*K1*K2*K3 + 232*K1*K3*K4 + 16*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 928*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 832*K2**4*K4 - 2592*K2**4 + 64*K2**3*K3*K5 - 576*K2**2*K3**2 - 136*K2**2*K4**2 + 1296*K2**2*K4 - 736*K2**2 + 136*K2*K3*K5 - 820*K3**2 - 172*K4**2 - 20*K5**2 + 2466
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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