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Flat knot 6.763

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,2,2,4,0,2,1,3,1,0,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.763']
Arrow polynomial of the knot is: -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.763', '6.1515', '6.1741', '6.1825']
Outer characteristic polynomial of the knot is: t^7+63t^5+37t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.763']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 64*K1**4*K2**2 + 768*K1**4*K2 - 4224*K1**4 + 224*K1**3*K2*K3 - 1248*K1**3*K3 + 64*K1**2*K2**3 - 3680*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 9736*K1**2*K2 - 1440*K1**2*K3**2 - 128*K1**2*K3*K5 - 256*K1**2*K4**2 - 6096*K1**2 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7896*K1*K2*K3 - 32*K1*K3**2*K5 + 2360*K1*K3*K4 + 336*K1*K4*K5 + 16*K1*K5*K6 - 304*K2**4 - 416*K2**2*K3**2 - 64*K2**2*K4**2 + 968*K2**2*K4 - 5104*K2**2 + 488*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 40*K3**2*K6 - 2792*K3**2 - 928*K4**2 - 176*K5**2 - 24*K6**2 + 5390
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.763']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3653', 'vk6.3750', 'vk6.3939', 'vk6.4036', 'vk6.4501', 'vk6.4598', 'vk6.5883', 'vk6.6012', 'vk6.7138', 'vk6.7311', 'vk6.7404', 'vk6.7932', 'vk6.8053', 'vk6.9362', 'vk6.17923', 'vk6.18018', 'vk6.18745', 'vk6.24458', 'vk6.24866', 'vk6.25327', 'vk6.37484', 'vk6.43885', 'vk6.44213', 'vk6.44516', 'vk6.48277', 'vk6.48342', 'vk6.50060', 'vk6.50170', 'vk6.50577', 'vk6.50642', 'vk6.55882', 'vk6.60721']
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 O1O2O3O4U2O5U6U5U4O6U1U3
R3 orbit {'O1O2O3O4U2O5U6U5U4O6U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U4O5U1U6U5O6U3
Gauss code of K* O1O2O3U1O4O5U4U6U5U3O6U2
Gauss code of -K* O1O2O3U2O4U1U5U4U6O5O6U3
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 -2 2 2 1 -2],[ 1 0 -1 2 2 1 -1],[ 2 1 0 2 1 0 1],[-2 -2 -2 0 1 1 -4],[-2 -2 -1 -1 0 0 -3],[-1 -1 0 -1 0 0 -1],[ 2 1 -1 4 3 1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 1 1 -2 -2 -4],[-2 -1 0 0 -2 -1 -3],[-1 -1 0 0 -1 0 -1],[ 1 2 2 1 0 -1 -1],[ 2 2 1 0 1 0 1],[ 2 4 3 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,-1,-1,2,2,4,0,2,1,3,1,0,1,1,1,-1]
Phi over symmetry [-2,-2,-1,1,2,2,-1,-1,2,2,4,0,2,1,3,1,0,1,1,1,-1]
Phi of -K [-2,-2,-1,1,2,2,-1,0,3,2,3,0,2,0,1,1,1,1,2,1,-1]
Phi of K* [-2,-2,-1,1,2,2,-1,1,1,1,3,2,1,0,2,1,2,3,0,0,-1]
Phi of -K* [-2,-2,-1,1,2,2,-1,1,1,3,4,1,0,1,2,1,2,2,0,-1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 19z+39
Enhanced Jones-Krushkal polynomial 19w^2z+39w
Inner characteristic polynomial t^6+45t^4+15t^2
Outer characteristic polynomial t^7+63t^5+37t^3+5t
Flat arrow polynomial -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -128*K1**6 - 64*K1**4*K2**2 + 768*K1**4*K2 - 4224*K1**4 + 224*K1**3*K2*K3 - 1248*K1**3*K3 + 64*K1**2*K2**3 - 3680*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 9736*K1**2*K2 - 1440*K1**2*K3**2 - 128*K1**2*K3*K5 - 256*K1**2*K4**2 - 6096*K1**2 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7896*K1*K2*K3 - 32*K1*K3**2*K5 + 2360*K1*K3*K4 + 336*K1*K4*K5 + 16*K1*K5*K6 - 304*K2**4 - 416*K2**2*K3**2 - 64*K2**2*K4**2 + 968*K2**2*K4 - 5104*K2**2 + 488*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 40*K3**2*K6 - 2792*K3**2 - 928*K4**2 - 176*K5**2 - 24*K6**2 + 5390
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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