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Flat knot 6.1718

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,2,0,0,2,2,-1,1,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1718']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']
Outer characteristic polynomial of the knot is: t^7+31t^5+70t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1718']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 1312*K1**4*K2 - 3600*K1**4 + 544*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1024*K1**3*K3 + 704*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 6656*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 11920*K1**2*K2 - 624*K1**2*K3**2 - 32*K1**2*K3*K5 - 384*K1**2*K4**2 - 8220*K1**2 + 256*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 256*K1*K2**2*K5 - 960*K1*K2*K3*K4 + 9392*K1*K2*K3 - 160*K1*K2*K4*K5 + 2456*K1*K3*K4 + 552*K1*K4*K5 - 32*K2**6 + 288*K2**4*K4 - 1360*K2**4 - 96*K2**3*K6 - 400*K2**2*K3**2 - 320*K2**2*K4**2 + 2696*K2**2*K4 - 6618*K2**2 + 656*K2*K3*K5 + 184*K2*K4*K6 - 3136*K3**2 - 1488*K4**2 - 196*K5**2 - 6*K6**2 + 6750
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1718']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17099', 'vk6.17342', 'vk6.20582', 'vk6.21990', 'vk6.23486', 'vk6.23825', 'vk6.28045', 'vk6.29503', 'vk6.35639', 'vk6.36080', 'vk6.39462', 'vk6.41661', 'vk6.43003', 'vk6.43315', 'vk6.46046', 'vk6.47712', 'vk6.55238', 'vk6.55490', 'vk6.57468', 'vk6.58630', 'vk6.59636', 'vk6.59984', 'vk6.62139', 'vk6.63101', 'vk6.65040', 'vk6.65239', 'vk6.66998', 'vk6.67862', 'vk6.68305', 'vk6.68455', 'vk6.69614', 'vk6.70306']
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 O1O2O3U1U4O5O6U5U3O4U2U6
R3 orbit {'O1O2O3U1U4O5O6U5U3O4U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O5U1U6O4O6U5U3
Gauss code of K* O1O2U3O4O5U6U4U2O6O3U1U5
Gauss code of -K* O1O2U3O4O5U1U5O3O6U4U2U6
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 0 1 0 -1 2],[ 2 0 2 1 1 0 2],[ 0 -2 0 1 -1 0 2],[-1 -1 -1 0 -1 0 1],[ 0 -1 1 1 0 -1 1],[ 1 0 0 0 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -1 -2],[-1 1 0 -1 -1 0 -1],[ 0 1 1 0 1 -1 -1],[ 0 2 1 -1 0 0 -2],[ 1 1 0 1 0 0 0],[ 2 2 1 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,2,1,2,1,1,0,1,-1,1,1,0,2,0]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,2,2,0,0,2,2,-1,1,0,0,1,1]
Phi of -K [-2,-1,0,0,1,2,1,0,1,2,2,1,0,2,2,1,0,0,0,1,0]
Phi of K* [-2,-1,0,0,1,2,0,0,1,2,2,0,0,2,2,-1,1,0,0,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,2,1,2,1,0,0,1,1,1,1,1,2,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+21t^4+38t^2+4
Outer characteristic polynomial t^7+31t^5+70t^3+14t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -192*K1**4*K2**2 + 1312*K1**4*K2 - 3600*K1**4 + 544*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1024*K1**3*K3 + 704*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 6656*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 11920*K1**2*K2 - 624*K1**2*K3**2 - 32*K1**2*K3*K5 - 384*K1**2*K4**2 - 8220*K1**2 + 256*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 256*K1*K2**2*K5 - 960*K1*K2*K3*K4 + 9392*K1*K2*K3 - 160*K1*K2*K4*K5 + 2456*K1*K3*K4 + 552*K1*K4*K5 - 32*K2**6 + 288*K2**4*K4 - 1360*K2**4 - 96*K2**3*K6 - 400*K2**2*K3**2 - 320*K2**2*K4**2 + 2696*K2**2*K4 - 6618*K2**2 + 656*K2*K3*K5 + 184*K2*K4*K6 - 3136*K3**2 - 1488*K4**2 - 196*K5**2 - 6*K6**2 + 6750
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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