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Flat knot 6.1120

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,0,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1120']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+42t^5+46t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1120']
2-strand cable arrow polynomial of the knot is: -2800*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 384*K1**3*K3 + 96*K1**2*K2**2*K4 - 3552*K1**2*K2**2 - 1696*K1**2*K2*K4 + 6752*K1**2*K2 - 304*K1**2*K3**2 - 464*K1**2*K4**2 - 4016*K1**2 - 512*K1*K2**2*K3 - 352*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6176*K1*K2*K3 + 2432*K1*K3*K4 + 552*K1*K4*K5 - 480*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 1520*K2**2*K4 - 3604*K2**2 + 320*K2*K3*K5 + 32*K2*K4*K6 - 2140*K3**2 - 1220*K4**2 - 204*K5**2 - 4*K6**2 + 3802
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1120']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13393', 'vk6.13484', 'vk6.13673', 'vk6.13773', 'vk6.14197', 'vk6.14438', 'vk6.15667', 'vk6.16121', 'vk6.16763', 'vk6.16779', 'vk6.16880', 'vk6.19047', 'vk6.19315', 'vk6.19608', 'vk6.23186', 'vk6.23261', 'vk6.25657', 'vk6.26503', 'vk6.33144', 'vk6.33201', 'vk6.33298', 'vk6.35173', 'vk6.35202', 'vk6.37751', 'vk6.42668', 'vk6.42685', 'vk6.42781', 'vk6.44735', 'vk6.53577', 'vk6.53705', 'vk6.54972', 'vk6.64615']
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 O1O2O3O4U3U2O5U6U4O6U1U5
R3 orbit {'O1O2O3O4U3U2O5U6U4O6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4O6U1U6O5U3U2
Gauss code of K* O1O2U3O4O5U4O6O3U6U2U1U5
Gauss code of -K* O1O2U3O4O3U1O5O6U4U6U5U2
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 -1 -1 2 2 -1],[ 1 0 -1 -1 3 2 0],[ 1 1 0 0 2 1 0],[ 1 1 0 0 1 1 0],[-2 -3 -2 -1 0 0 -2],[-2 -2 -1 -1 0 0 -2],[ 1 0 0 0 2 2 0]]
Primitive based matrix [[ 0 2 2 -1 -1 -1 -1],[-2 0 0 -1 -1 -2 -2],[-2 0 0 -1 -2 -2 -3],[ 1 1 1 0 0 0 1],[ 1 1 2 0 0 0 1],[ 1 2 2 0 0 0 0],[ 1 2 3 -1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,1,1,1,1,0,1,1,2,2,1,2,2,3,0,0,-1,0,-1,0]
Phi over symmetry [-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,0,-1,-1,0,0,0]
Phi of -K [-1,-1,-1,-1,2,2,-1,0,0,1,2,0,1,0,1,0,1,1,2,2,0]
Phi of K* [-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,0,-1,-1,0,0,0]
Phi of -K* [-1,-1,-1,-1,2,2,-1,-1,0,2,3,0,0,1,1,0,1,2,2,2,0]
Symmetry type of based matrix c
u-polynomial -2t^2+4t
Normalized Jones-Krushkal polynomial 8z^2+27z+23
Enhanced Jones-Krushkal polynomial -4w^4z^2+12w^3z^2+27w^2z+23w
Inner characteristic polynomial t^6+30t^4+28t^2+4
Outer characteristic polynomial t^7+42t^5+46t^3+9t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial -2800*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 384*K1**3*K3 + 96*K1**2*K2**2*K4 - 3552*K1**2*K2**2 - 1696*K1**2*K2*K4 + 6752*K1**2*K2 - 304*K1**2*K3**2 - 464*K1**2*K4**2 - 4016*K1**2 - 512*K1*K2**2*K3 - 352*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6176*K1*K2*K3 + 2432*K1*K3*K4 + 552*K1*K4*K5 - 480*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 1520*K2**2*K4 - 3604*K2**2 + 320*K2*K3*K5 + 32*K2*K4*K6 - 2140*K3**2 - 1220*K4**2 - 204*K5**2 - 4*K6**2 + 3802
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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