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Flat knot 6.1088

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,2,2,2,4,1,-1,0,0,0,0,1,1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1088']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+42t^5+54t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1088']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 704*K1**4*K2 - 2656*K1**4 + 480*K1**3*K2*K3 - 832*K1**3*K3 - 192*K1**2*K2**4 + 480*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 3792*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 7264*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 4348*K1**2 + 160*K1*K2**3*K3 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5056*K1*K2*K3 + 832*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 368*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 360*K2**2*K4 - 3176*K2**2 + 48*K2*K3*K5 - 1472*K3**2 - 220*K4**2 - 4*K5**2 + 3410
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1088']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11518', 'vk6.11851', 'vk6.12868', 'vk6.13177', 'vk6.20352', 'vk6.21694', 'vk6.27655', 'vk6.29200', 'vk6.31293', 'vk6.31690', 'vk6.32451', 'vk6.32868', 'vk6.39092', 'vk6.41348', 'vk6.45848', 'vk6.47513', 'vk6.52301', 'vk6.52567', 'vk6.53145', 'vk6.53451', 'vk6.57223', 'vk6.58448', 'vk6.61836', 'vk6.62972', 'vk6.63810', 'vk6.63944', 'vk6.64256', 'vk6.64454', 'vk6.66838', 'vk6.67707', 'vk6.69477', 'vk6.70200']
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 O1O2O3O4U1U3O5U4U2O6U5U6
R3 orbit {'O1O2O3O4U1U3O5U4U2O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6O5U3U1O6U2U4
Gauss code of K* O1O2U3O4O5U6O3O6U1U5U2U4
Gauss code of -K* O1O2U1O3O4U2O5O6U4U5U3U6
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 -3 0 0 1 1 1],[ 3 0 3 1 2 2 0],[ 0 -3 0 -1 1 2 1],[ 0 -1 1 0 1 1 0],[-1 -2 -1 -1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-1 0 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 1 -1 -1 -2],[-1 -1 0 1 -1 -2 -2],[-1 -1 -1 0 0 -1 0],[ 0 1 1 0 0 1 -1],[ 0 1 2 1 -1 0 -3],[ 3 2 2 0 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,-1,1,1,2,-1,1,2,2,0,1,0,-1,1,3]
Phi over symmetry [-3,0,0,1,1,1,0,2,2,2,4,1,-1,0,0,0,0,1,1,-1,-1]
Phi of -K [-3,0,0,1,1,1,0,2,2,2,4,1,-1,0,0,0,0,1,1,-1,-1]
Phi of K* [-1,-1,-1,0,0,3,-1,-1,0,1,4,-1,-1,0,2,0,0,2,-1,0,2]
Phi of -K* [-3,0,0,1,1,1,1,3,0,2,2,1,0,1,1,1,1,2,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+30t^4+8t^2
Outer characteristic polynomial t^7+42t^5+54t^3+4t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -64*K1**6 + 704*K1**4*K2 - 2656*K1**4 + 480*K1**3*K2*K3 - 832*K1**3*K3 - 192*K1**2*K2**4 + 480*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 3792*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 7264*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 4348*K1**2 + 160*K1*K2**3*K3 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5056*K1*K2*K3 + 832*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 368*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 360*K2**2*K4 - 3176*K2**2 + 48*K2*K3*K5 - 1472*K3**2 - 220*K4**2 - 4*K5**2 + 3410
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{6}, {3, 5}, {1, 4}, {2}]]
If K is slice False
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