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Flat knot 6.128

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,-1,1,1,2,3,1,1,1,1,-1,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.128', '7.10420']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+53t^5+72t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.128']
2-strand cable arrow polynomial of the knot is: -576*K1**4 + 384*K1**3*K3*K4 - 384*K1**3*K3 - 720*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 224*K1**2*K2*K4 + 1640*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K3*K5 - 752*K1**2*K4**2 - 32*K1**2*K4*K6 - 1476*K1**2 + 384*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 128*K1*K2**2*K3 + 32*K1*K2*K3*K4**2 - 1024*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 2792*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K3*K4*K6 + 2008*K1*K3*K4 + 664*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 160*K2**4*K4 - 344*K2**4 + 32*K2**3*K4*K6 - 848*K2**2*K3**2 + 32*K2**2*K4**3 - 840*K2**2*K4**2 + 1048*K2**2*K4 - 8*K2**2*K6**2 - 1154*K2**2 - 192*K2*K3**2*K4 + 672*K2*K3*K5 - 32*K2*K4**2*K6 + 472*K2*K4*K6 + 8*K2*K6*K8 - 16*K3**2*K4**2 - 1120*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 828*K4**2 - 120*K5**2 - 30*K6**2 - 4*K7**2 - 2*K8**2 + 1516
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.128']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.492', 'vk6.564', 'vk6.632', 'vk6.975', 'vk6.1066', 'vk6.1142', 'vk6.1660', 'vk6.1777', 'vk6.1859', 'vk6.2151', 'vk6.2244', 'vk6.2324', 'vk6.2587', 'vk6.2906', 'vk6.3073', 'vk6.3180', 'vk6.12064', 'vk6.13055', 'vk6.20511', 'vk6.21096', 'vk6.21888', 'vk6.22526', 'vk6.27943', 'vk6.28540', 'vk6.29425', 'vk6.32714', 'vk6.39356', 'vk6.41531', 'vk6.46818', 'vk6.46904', 'vk6.53292', 'vk6.57374']
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 O1O2O3O4O5O6U3U6U5U4U1U2
R3 orbit {'O1O2O3O4O5O6U3U6U5U4U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U6U3U2U1U4
Gauss code of K* O1O2O3O4O5O6U5U6U1U4U3U2
Gauss code of -K* O1O2O3O4O5O6U5U4U3U6U1U2
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 -3 1 1 1],[ 1 0 1 -3 1 1 1],[-1 -1 0 -3 1 1 1],[ 3 3 3 0 3 2 1],[-1 -1 -1 -3 0 0 0],[-1 -1 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 1 -1 -3],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 0 -1 -2],[-1 -1 0 0 0 -1 -3],[ 1 1 1 1 1 0 -3],[ 3 3 1 2 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,-1,1,3,0,0,1,1,0,1,2,1,3,3]
Phi over symmetry [-3,-1,1,1,1,1,-1,1,1,2,3,1,1,1,1,-1,-1,-1,0,0,0]
Phi of -K [-3,-1,1,1,1,1,-1,1,1,2,3,1,1,1,1,-1,-1,-1,0,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,0,0,1,1,1,1,1,1,0,1,2,1,3,-1]
Phi of -K* [-3,-1,1,1,1,1,3,1,2,3,3,1,1,1,1,0,-1,0,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial t^6+39t^4+26t^2
Outer characteristic polynomial t^7+53t^5+72t^3+6t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial -576*K1**4 + 384*K1**3*K3*K4 - 384*K1**3*K3 - 720*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 224*K1**2*K2*K4 + 1640*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K3*K5 - 752*K1**2*K4**2 - 32*K1**2*K4*K6 - 1476*K1**2 + 384*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 128*K1*K2**2*K3 + 32*K1*K2*K3*K4**2 - 1024*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 2792*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K3*K4*K6 + 2008*K1*K3*K4 + 664*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 160*K2**4*K4 - 344*K2**4 + 32*K2**3*K4*K6 - 848*K2**2*K3**2 + 32*K2**2*K4**3 - 840*K2**2*K4**2 + 1048*K2**2*K4 - 8*K2**2*K6**2 - 1154*K2**2 - 192*K2*K3**2*K4 + 672*K2*K3*K5 - 32*K2*K4**2*K6 + 472*K2*K4*K6 + 8*K2*K6*K8 - 16*K3**2*K4**2 - 1120*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 828*K4**2 - 120*K5**2 - 30*K6**2 - 4*K7**2 - 2*K8**2 + 1516
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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