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Flat knot 6.1509

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,2,3,3,-1,2,1,1,2,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1509']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1209', '6.1245', '6.1509', '6.1541', '6.1704', '6.1778', '6.1914']
Outer characteristic polynomial of the knot is: t^7+30t^5+67t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1509']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 512*K1**4*K2 - 2128*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 992*K1**3*K3 + 256*K1**2*K2**3 - 2400*K1**2*K2**2 + 224*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 6120*K1**2*K2 - 1552*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 4536*K1**2 + 224*K1*K2**3*K3 - 992*K1*K2**2*K3 - 128*K1*K2**2*K5 + 160*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6040*K1*K2*K3 - 32*K1*K3**2*K5 + 1808*K1*K3*K4 + 224*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 448*K2**4 - 32*K2**3*K6 - 576*K2**2*K3**2 - 32*K2**2*K4**2 + 904*K2**2*K4 - 3634*K2**2 - 32*K2*K3**2*K4 + 520*K2*K3*K5 + 48*K2*K4*K6 - 128*K3**4 + 72*K3**2*K6 - 2160*K3**2 - 688*K4**2 - 152*K5**2 - 14*K6**2 + 3862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1509']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4821', 'vk6.5166', 'vk6.6385', 'vk6.6818', 'vk6.8350', 'vk6.8784', 'vk6.9722', 'vk6.10027', 'vk6.11614', 'vk6.11967', 'vk6.12960', 'vk6.20465', 'vk6.20739', 'vk6.21819', 'vk6.27850', 'vk6.29359', 'vk6.31417', 'vk6.32595', 'vk6.39286', 'vk6.39779', 'vk6.41466', 'vk6.46343', 'vk6.47588', 'vk6.47920', 'vk6.49048', 'vk6.49876', 'vk6.51308', 'vk6.51527', 'vk6.53225', 'vk6.57336', 'vk6.62023', 'vk6.64306']
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 O1O2O3U2O4U1U3O5O6U5U4U6
R3 orbit {'O1O2O3U2O4U1U3O5O6U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U6O4O6U1U3O5U2
Gauss code of K* O1O2O3U4U5U6O5U2O4O6U1U3
Gauss code of -K* O1O2O3U1U3O4O5U2O6U4U6U5
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 -1 1 1 -1 2],[ 2 0 0 2 2 0 1],[ 1 0 0 1 1 0 0],[-1 -2 -1 0 1 0 1],[-1 -2 -1 -1 0 0 2],[ 1 0 0 0 0 0 1],[-2 -1 0 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -2 0 -1 -1],[-1 1 0 1 -1 0 -2],[-1 2 -1 0 -1 0 -2],[ 1 0 1 1 0 0 0],[ 1 1 0 0 0 0 0],[ 2 1 2 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,2,0,1,1,-1,1,0,2,1,0,2,0,0,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,2,3,3,-1,2,1,1,2,1,1,0,1,1]
Phi of -K [-2,-1,-1,1,1,2,1,1,1,1,3,0,1,1,3,2,2,2,-1,0,-1]
Phi of K* [-2,-1,-1,1,1,2,-1,0,2,3,3,-1,2,1,1,2,1,1,0,1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,0,2,2,1,0,0,0,1,1,1,0,-1,2,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+18t^4+23t^2
Outer characteristic polynomial t^7+30t^5+67t^3+4t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -128*K1**6 + 512*K1**4*K2 - 2128*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 992*K1**3*K3 + 256*K1**2*K2**3 - 2400*K1**2*K2**2 + 224*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 6120*K1**2*K2 - 1552*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 4536*K1**2 + 224*K1*K2**3*K3 - 992*K1*K2**2*K3 - 128*K1*K2**2*K5 + 160*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6040*K1*K2*K3 - 32*K1*K3**2*K5 + 1808*K1*K3*K4 + 224*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 448*K2**4 - 32*K2**3*K6 - 576*K2**2*K3**2 - 32*K2**2*K4**2 + 904*K2**2*K4 - 3634*K2**2 - 32*K2*K3**2*K4 + 520*K2*K3*K5 + 48*K2*K4*K6 - 128*K3**4 + 72*K3**2*K6 - 2160*K3**2 - 688*K4**2 - 152*K5**2 - 14*K6**2 + 3862
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice True
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