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Flat knot 6.1198

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-2,0,1,2,2,1,0,0,2,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1198']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.133', '6.517', '6.545', '6.1198', '6.1251', '6.1906']
Outer characteristic polynomial of the knot is: t^7+33t^5+75t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1198']
2-strand cable arrow polynomial of the knot is: 1920*K1**2*K2**5 - 4096*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3680*K1**2*K2**3 - 3968*K1**2*K2**2 - 288*K1**2*K2*K4 + 1968*K1**2*K2 - 1088*K1**2 + 512*K1*K2**5*K3 - 1024*K1*K2**4*K3 - 256*K1*K2**4*K5 - 256*K1*K2**3*K3*K4 + 2496*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 1920*K1*K2*K3 + 96*K1*K3*K4 + 96*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1984*K2**6 - 128*K2**5*K6 - 256*K2**4*K3**2 - 64*K2**4*K4**2 + 1536*K2**4*K4 - 1376*K2**4 + 288*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 416*K2**2*K3**2 - 224*K2**2*K4**2 + 1072*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 + 384*K2**2 + 224*K2*K3*K5 + 56*K2*K4*K6 - 272*K3**2 - 146*K4**2 - 64*K5**2 - 8*K6**2 + 760
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1198']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4765', 'vk6.5100', 'vk6.6330', 'vk6.6760', 'vk6.8288', 'vk6.8740', 'vk6.9660', 'vk6.9971', 'vk6.20714', 'vk6.22158', 'vk6.28266', 'vk6.29690', 'vk6.39725', 'vk6.41975', 'vk6.46287', 'vk6.47876', 'vk6.48797', 'vk6.49012', 'vk6.49628', 'vk6.49830', 'vk6.50829', 'vk6.51049', 'vk6.51302', 'vk6.51495', 'vk6.57649', 'vk6.58795', 'vk6.62320', 'vk6.63260', 'vk6.67113', 'vk6.67978', 'vk6.69707', 'vk6.70388']
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 O1O2O3O4U2U3U4O5O6U1U6U5
R3 orbit {'O1O2O3O4U2U3U4O5O6U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U4O6O5U1U2U3
Gauss code of K* O1O2O3U4U5O6O5O4U6U1U2U3
Gauss code of -K* O1O2O3U2U1O4O5O6U4U5U6U3
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 0 2 1 1],[ 2 0 -2 0 2 2 1],[ 2 2 0 1 2 0 0],[ 0 0 -1 0 1 0 0],[-2 -2 -2 -1 0 0 0],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[ 0 1 0 0 0 -1 0],[ 2 2 0 0 1 0 2],[ 2 2 1 2 0 -2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,0,0,1,2,2,0,0,0,1,0,0,2,1,0,-2]
Phi over symmetry [-2,-2,0,1,1,2,-2,0,1,2,2,1,0,0,2,0,0,1,0,0,0]
Phi of -K [-2,-2,0,1,1,2,-2,1,3,3,2,2,1,2,2,1,1,1,0,1,1]
Phi of K* [-2,-1,-1,0,2,2,1,1,1,2,2,0,1,1,3,1,2,3,2,1,-2]
Phi of -K* [-2,-2,0,1,1,2,-2,0,1,2,2,1,0,0,2,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+8z+5
Enhanced Jones-Krushkal polynomial -6w^4z^2+9w^3z^2-10w^3z+18w^2z+5w
Inner characteristic polynomial t^6+19t^4+30t^2
Outer characteristic polynomial t^7+33t^5+75t^3+6t
Flat arrow polynomial -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
2-strand cable arrow polynomial 1920*K1**2*K2**5 - 4096*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3680*K1**2*K2**3 - 3968*K1**2*K2**2 - 288*K1**2*K2*K4 + 1968*K1**2*K2 - 1088*K1**2 + 512*K1*K2**5*K3 - 1024*K1*K2**4*K3 - 256*K1*K2**4*K5 - 256*K1*K2**3*K3*K4 + 2496*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 1920*K1*K2*K3 + 96*K1*K3*K4 + 96*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1984*K2**6 - 128*K2**5*K6 - 256*K2**4*K3**2 - 64*K2**4*K4**2 + 1536*K2**4*K4 - 1376*K2**4 + 288*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 416*K2**2*K3**2 - 224*K2**2*K4**2 + 1072*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 + 384*K2**2 + 224*K2*K3*K5 + 56*K2*K4*K6 - 272*K3**2 - 146*K4**2 - 64*K5**2 - 8*K6**2 + 760
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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