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Flat knot 6.1199

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,0,1,0,2,2,0,0,1,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.1199']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 + 4*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.544', '6.1199']
Outer characteristic polynomial of the knot is: t^7+33t^5+91t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1199']
2-strand cable arrow polynomial of the knot is: 1920*K1**2*K2**5 - 5248*K1**2*K2**4 - 640*K1**2*K2**3*K4 + 4320*K1**2*K2**3 - 5600*K1**2*K2**2 - 384*K1**2*K2*K4 + 2800*K1**2*K2 - 1472*K1**2 + 896*K1*K2**5*K3 - 1024*K1*K2**4*K3 - 256*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 4288*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 - 192*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3232*K1*K2*K3 + 128*K1*K3*K4 - 128*K2**8 + 256*K2**6*K4 - 2304*K2**6 - 896*K2**4*K3**2 - 192*K2**4*K4**2 + 1696*K2**4*K4 - 1888*K2**4 + 480*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1152*K2**2*K3**2 - 240*K2**2*K4**2 + 1296*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 + 496*K2**2 + 384*K2*K3*K5 + 56*K2*K4*K6 - 496*K3**2 - 130*K4**2 - 32*K5**2 - 8*K6**2 + 1064
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1199']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3245', 'vk6.3262', 'vk6.3271', 'vk6.3361', 'vk6.3398', 'vk6.3407', 'vk6.3457', 'vk6.3508', 'vk6.17649', 'vk6.17654', 'vk6.17656', 'vk6.24202', 'vk6.24208', 'vk6.24217', 'vk6.24222', 'vk6.36462', 'vk6.36477', 'vk6.36484', 'vk6.43559', 'vk6.43565', 'vk6.43574', 'vk6.43579', 'vk6.48099', 'vk6.48102', 'vk6.48130', 'vk6.48135', 'vk6.48166', 'vk6.48175', 'vk6.48184', 'vk6.48189', 'vk6.60247', 'vk6.68537']
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 O1O2O3O4U2U3U4O5O6U5U1U6
R3 orbit {'O1O2O3O4U2U3U4O5O6U5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U6O5O6U1U2U3
Gauss code of K* O1O2O3U4U5O4O6O5U6U1U2U3
Gauss code of -K* O1O2O3U1U3O4O5O6U4U5U6U2
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 0 2 -1 2],[ 1 0 -2 0 2 0 2],[ 2 2 0 1 2 0 0],[ 0 0 -1 0 1 0 0],[-2 -2 -2 -1 0 0 0],[ 1 0 0 0 0 0 1],[-2 -2 0 0 0 -1 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 0 0 -1 -2 0],[-2 0 0 -1 0 -2 -2],[ 0 0 1 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 2 2 0 0 0 -2],[ 2 0 2 1 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,0,0,1,2,0,1,0,2,2,0,0,1,0,0,2]
Phi over symmetry [-2,-2,0,1,1,2,0,0,1,2,0,1,0,2,2,0,0,1,0,0,2]
Phi of -K [-2,-1,-1,0,2,2,-1,1,1,2,4,0,1,1,1,1,3,2,1,2,0]
Phi of K* [-2,-2,0,1,1,2,0,1,1,3,2,2,1,2,4,1,1,1,0,-1,1]
Phi of -K* [-2,-1,-1,0,2,2,0,2,1,0,2,0,0,1,0,0,2,2,0,1,0]
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+38t^2
Outer characteristic polynomial t^7+33t^5+91t^3+6t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 + 4*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 - K2
2-strand cable arrow polynomial 1920*K1**2*K2**5 - 5248*K1**2*K2**4 - 640*K1**2*K2**3*K4 + 4320*K1**2*K2**3 - 5600*K1**2*K2**2 - 384*K1**2*K2*K4 + 2800*K1**2*K2 - 1472*K1**2 + 896*K1*K2**5*K3 - 1024*K1*K2**4*K3 - 256*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 4288*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 - 192*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3232*K1*K2*K3 + 128*K1*K3*K4 - 128*K2**8 + 256*K2**6*K4 - 2304*K2**6 - 896*K2**4*K3**2 - 192*K2**4*K4**2 + 1696*K2**4*K4 - 1888*K2**4 + 480*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1152*K2**2*K3**2 - 240*K2**2*K4**2 + 1296*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 + 496*K2**2 + 384*K2*K3*K5 + 56*K2*K4*K6 - 496*K3**2 - 130*K4**2 - 32*K5**2 - 8*K6**2 + 1064
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}, {2, 5}, {1, 3}], [{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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