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Flat knot 6.926

Min(phi) over symmetries of the knot is: [-3,-1,2,2,1,1,2,2,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.926']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^5+45t^3+51t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.926']
2-strand cable arrow polynomial of the knot is: 1056*K1**4*K2 - 1728*K1**4 - 128*K1**3*K2**2*K3 + 800*K1**3*K2*K3 - 384*K1**3*K3 + 128*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3088*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 4832*K1**2*K2 - 640*K1**2*K3**2 - 3804*K1**2 + 160*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 160*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 5256*K1*K2*K3 - 192*K1*K2*K4*K5 + 1704*K1*K3*K4 + 376*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**4*K4**2 + 64*K2**4*K4 - 256*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 544*K2**2*K3**2 - 32*K2**2*K3*K7 - 360*K2**2*K4**2 - 32*K2**2*K4*K8 + 1760*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 4406*K2**2 - 160*K2*K3**2*K4 + 1184*K2*K3*K5 + 696*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 2432*K3**2 - 1328*K4**2 - 448*K5**2 - 202*K6**2 - 4*K7**2 - 2*K8**2 + 4128
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.926']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11200', 'vk6.11211', 'vk6.11215', 'vk6.12397', 'vk6.12401', 'vk6.12412', 'vk6.12416', 'vk6.14509', 'vk6.14510', 'vk6.15730', 'vk6.15731', 'vk6.16157', 'vk6.16158', 'vk6.30791', 'vk6.30813', 'vk6.30817', 'vk6.32001', 'vk6.32005', 'vk6.34080', 'vk6.34196', 'vk6.34478', 'vk6.34517', 'vk6.51936', 'vk6.51959', 'vk6.51963', 'vk6.54151', 'vk6.54152', 'vk6.54343', 'vk6.54344', 'vk6.63610', 'vk6.63621', 'vk6.63625']
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 O1O2O3O4U3U5O6O5U2U1U6U4
R3 orbit {'O1O2O3O4U3U5O6O5U2U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4U3O6O5U6U2
Gauss code of K* O1O2O3O4U2U1U5U4O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U1U6U4U3
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 -2 -1 3 1 1],[ 2 0 0 0 4 2 1],[ 2 0 0 0 3 2 0],[ 1 0 0 0 1 1 0],[-3 -4 -3 -1 0 -2 -1],[-1 -2 -2 -1 2 0 1],[-1 -1 0 0 1 -1 0]]
Primitive based matrix [[ 0 3 1 -2 -2],[-3 0 -1 -3 -4],[-1 1 0 0 -1],[ 2 3 0 0 0],[ 2 4 1 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-1,2,2,1,3,4,0,1,0]
Phi over symmetry [-3,-1,2,2,1,1,2,2,3,0]
Phi of -K [-2,-2,1,3,0,2,1,3,2,1]
Phi of K* [-3,-1,2,2,1,1,2,2,3,0]
Phi of -K* [-2,-2,1,3,0,0,3,1,4,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^4+27t^2+9
Outer characteristic polynomial t^5+45t^3+51t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial 1056*K1**4*K2 - 1728*K1**4 - 128*K1**3*K2**2*K3 + 800*K1**3*K2*K3 - 384*K1**3*K3 + 128*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3088*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 4832*K1**2*K2 - 640*K1**2*K3**2 - 3804*K1**2 + 160*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 160*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 5256*K1*K2*K3 - 192*K1*K2*K4*K5 + 1704*K1*K3*K4 + 376*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**4*K4**2 + 64*K2**4*K4 - 256*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 544*K2**2*K3**2 - 32*K2**2*K3*K7 - 360*K2**2*K4**2 - 32*K2**2*K4*K8 + 1760*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 4406*K2**2 - 160*K2*K3**2*K4 + 1184*K2*K3*K5 + 696*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 2432*K3**2 - 1328*K4**2 - 448*K5**2 - 202*K6**2 - 4*K7**2 - 2*K8**2 + 4128
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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