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Flat knot 6.74

Min(phi) over symmetries of the knot is: [-4,-2,-2,2,3,3,0,1,4,3,4,0,2,1,2,3,2,3,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.74']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500']
Outer characteristic polynomial of the knot is: t^7+123t^5+88t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.74']
2-strand cable arrow polynomial of the knot is: 480*K1**2*K2**3 - 1376*K1**2*K2**2 - 352*K1**2*K2*K4 + 1416*K1**2*K2 - 192*K1**2*K3**2 - 64*K1**2*K3*K5 - 1304*K1**2 + 192*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 1648*K1*K2*K3 + 608*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 192*K2**2*K3**2 - 336*K2**2*K4**2 + 856*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 824*K2**2 + 296*K2*K3*K5 + 128*K2*K4*K6 - 520*K3**2 - 386*K4**2 - 96*K5**2 - 8*K6**2 + 1024
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.74']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73189', 'vk6.73203', 'vk6.73627', 'vk6.74166', 'vk6.74314', 'vk6.74408', 'vk6.74747', 'vk6.74960', 'vk6.75019', 'vk6.75102', 'vk6.75561', 'vk6.75587', 'vk6.76279', 'vk6.76306', 'vk6.76526', 'vk6.76590', 'vk6.76819', 'vk6.76935', 'vk6.78037', 'vk6.78527', 'vk6.78553', 'vk6.79181', 'vk6.79196', 'vk6.79362', 'vk6.79648', 'vk6.79786', 'vk6.79855', 'vk6.80664', 'vk6.80678', 'vk6.80820', 'vk6.84185', 'vk6.84294', 'vk6.85269', 'vk6.85883', 'vk6.86197', 'vk6.87490', 'vk6.87703', 'vk6.88163', 'vk6.88375', 'vk6.88579']
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 O1O2O3O4O5O6U2U3U6U1U5U4
R3 orbit {'O1O2O3O4O5U1O6U3U2U6U5U4', 'O1O2O3O4O5O6U2U3U6U1U5U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U3U2U6U1U4U5
Gauss code of K* O1O2O3O4O5O6U4U1U2U6U5U3
Gauss code of -K* O1O2O3O4O5O6U4U2U1U5U6U3
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 -4 -2 3 3 2],[ 2 0 -2 0 4 3 2],[ 4 2 0 1 4 3 2],[ 2 0 -1 0 3 2 1],[-3 -4 -4 -3 0 0 0],[-3 -3 -3 -2 0 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 3 2 -2 -2 -4],[-3 0 0 0 -2 -3 -3],[-3 0 0 0 -3 -4 -4],[-2 0 0 0 -1 -2 -2],[ 2 2 3 1 0 0 -1],[ 2 3 4 2 0 0 -2],[ 4 3 4 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-2,2,2,4,0,0,2,3,3,0,3,4,4,1,2,2,0,1,2]
Phi over symmetry [-4,-2,-2,2,3,3,0,1,4,3,4,0,2,1,2,3,2,3,1,1,0]
Phi of -K [-4,-2,-2,2,3,3,0,1,4,3,4,0,2,1,2,3,2,3,1,1,0]
Phi of K* [-3,-3,-2,2,2,4,0,1,1,2,3,1,2,3,4,2,3,4,0,0,1]
Phi of -K* [-4,-2,-2,2,3,3,1,2,2,3,4,0,1,2,3,2,3,4,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+77t^4+17t^2
Outer characteristic polynomial t^7+123t^5+88t^3+3t
Flat arrow polynomial 4*K1**2*K2 - 2*K1*K3 - K2
2-strand cable arrow polynomial 480*K1**2*K2**3 - 1376*K1**2*K2**2 - 352*K1**2*K2*K4 + 1416*K1**2*K2 - 192*K1**2*K3**2 - 64*K1**2*K3*K5 - 1304*K1**2 + 192*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 1648*K1*K2*K3 + 608*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 192*K2**2*K3**2 - 336*K2**2*K4**2 + 856*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 824*K2**2 + 296*K2*K3*K5 + 128*K2*K4*K6 - 520*K3**2 - 386*K4**2 - 96*K5**2 - 8*K6**2 + 1024
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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