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Flat knot 6.685

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,1,2,4,0,0,0,1,1,2,2,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.685']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+54t^5+47t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.685']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 608*K1**4*K2 - 3200*K1**4 + 384*K1**3*K2*K3 + 96*K1**3*K3*K4 - 864*K1**3*K3 + 256*K1**2*K2**3 - 3344*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 7064*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K3*K5 - 128*K1**2*K4**2 - 3752*K1**2 + 128*K1*K2**3*K3 - 224*K1*K2**2*K3 + 64*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 5488*K1*K2*K3 - 32*K1*K3**2*K5 + 1256*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 448*K2**4 - 336*K2**2*K3**2 - 56*K2**2*K4**2 + 536*K2**2*K4 - 3060*K2**2 + 264*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 48*K3**2*K6 - 1540*K3**2 - 312*K4**2 - 44*K5**2 - 12*K6**2 + 3294
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.685']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4833', 'vk6.5178', 'vk6.6401', 'vk6.6834', 'vk6.8366', 'vk6.8796', 'vk6.9734', 'vk6.10039', 'vk6.11615', 'vk6.11966', 'vk6.12961', 'vk6.20453', 'vk6.20743', 'vk6.21807', 'vk6.27838', 'vk6.29347', 'vk6.31418', 'vk6.32596', 'vk6.39270', 'vk6.39775', 'vk6.41450', 'vk6.46339', 'vk6.47576', 'vk6.47916', 'vk6.49060', 'vk6.49892', 'vk6.51320', 'vk6.51539', 'vk6.53226', 'vk6.57324', 'vk6.62011', 'vk6.64307']
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 O1O2O3O4U3O5U2U4O6U1U6U5
R3 orbit {'O1O2O3O4U3O5U2U4O6U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U4O6U1U3O5U2
Gauss code of K* O1O2O3U1U4U5U6O5U3O4O6U2
Gauss code of -K* O1O2O3U2O4O5U1O6U4U6U5U3
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 1 3 1],[ 2 0 -1 -1 2 4 1],[ 2 1 0 0 2 2 0],[ 1 1 0 0 1 1 0],[-1 -2 -2 -1 0 1 0],[-3 -4 -2 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -1 -1 -2 -4],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -2 -2],[ 1 1 0 1 0 0 1],[ 2 2 0 2 0 0 1],[ 2 4 1 2 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,1,1,2,4,0,0,0,1,1,2,2,0,-1,-1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,1,2,4,0,0,0,1,1,2,2,0,-1,-1]
Phi of -K [-2,-2,-1,1,1,3,-1,1,1,3,3,2,1,2,1,1,2,3,0,1,2]
Phi of K* [-3,-1,-1,1,2,2,1,2,3,1,3,0,1,1,1,2,2,3,2,1,-1]
Phi of -K* [-2,-2,-1,1,1,3,-1,-1,1,2,4,0,0,2,2,0,1,1,0,0,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+34t^4+23t^2
Outer characteristic polynomial t^7+54t^5+47t^3+4t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -128*K1**6 + 608*K1**4*K2 - 3200*K1**4 + 384*K1**3*K2*K3 + 96*K1**3*K3*K4 - 864*K1**3*K3 + 256*K1**2*K2**3 - 3344*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 7064*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K3*K5 - 128*K1**2*K4**2 - 3752*K1**2 + 128*K1*K2**3*K3 - 224*K1*K2**2*K3 + 64*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 5488*K1*K2*K3 - 32*K1*K3**2*K5 + 1256*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 448*K2**4 - 336*K2**2*K3**2 - 56*K2**2*K4**2 + 536*K2**2*K4 - 3060*K2**2 + 264*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 48*K3**2*K6 - 1540*K3**2 - 312*K4**2 - 44*K5**2 - 12*K6**2 + 3294
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}]]
If K is slice False
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