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Flat knot 6.183

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,3,1,2,4,1,1,1,1,1,0,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.183']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.73', '6.171', '6.183']
Outer characteristic polynomial of the knot is: t^7+61t^5+103t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.183']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 640*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 512*K1**2*K2**4 + 512*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 1936*K1**2*K2**2 + 1728*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K4**2 - 992*K1**2 + 960*K1*K2**3*K3 + 32*K1*K2*K3**3 + 1568*K1*K2*K3 + 152*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 752*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 608*K2**2*K3**2 - 80*K2**2*K4**2 + 328*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 364*K2**2 + 176*K2*K3*K5 + 24*K2*K4*K6 - 464*K3**2 - 146*K4**2 - 24*K5**2 - 4*K6**2 + 1000
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.183']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4674', 'vk6.4965', 'vk6.6140', 'vk6.6625', 'vk6.8147', 'vk6.8551', 'vk6.9523', 'vk6.9878', 'vk6.17672', 'vk6.17721', 'vk6.22144', 'vk6.24239', 'vk6.28237', 'vk6.29662', 'vk6.29908', 'vk6.29943', 'vk6.30006', 'vk6.30067', 'vk6.36505', 'vk6.39693', 'vk6.41934', 'vk6.43608', 'vk6.46265', 'vk6.47872', 'vk6.48714', 'vk6.48925', 'vk6.49490', 'vk6.49703', 'vk6.51615', 'vk6.51646', 'vk6.51691', 'vk6.51718', 'vk6.55706', 'vk6.58789', 'vk6.60280', 'vk6.63248', 'vk6.63353', 'vk6.63397', 'vk6.65410', 'vk6.68552']
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 O1O2O3O4O5U1O6U5U6U4U2U3
R3 orbit {'O1O2O3O4O5U1O6U5U6U4U2U3', 'O1O2O3O4O5U1U4O6U5U6U2U3'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U3U4U2U6U1O6U5
Gauss code of K* O1O2O3O4O5U6U4U5U3U1O6U2
Gauss code of -K* O1O2O3O4O5U4O6U5U3U1U2U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 0 2 1 0 1],[ 4 0 3 4 2 1 1],[ 0 -3 0 1 0 -1 1],[-2 -4 -1 0 0 -1 1],[-1 -2 0 0 0 -1 1],[ 0 -1 1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 0 -4],[-2 0 1 0 -1 -1 -4],[-1 -1 0 -1 -1 -1 -1],[-1 0 1 0 0 -1 -2],[ 0 1 1 0 0 -1 -3],[ 0 1 1 1 1 0 -1],[ 4 4 1 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,0,4,-1,0,1,1,4,1,1,1,1,0,1,2,1,3,1]
Phi over symmetry [-4,0,0,1,1,2,1,3,1,2,4,1,1,1,1,1,0,1,-1,-1,0]
Phi of -K [-4,0,0,1,1,2,1,3,3,4,2,1,1,0,1,0,0,1,-1,1,2]
Phi of K* [-2,-1,-1,0,0,4,1,2,1,1,2,1,0,1,3,0,0,4,1,3,1]
Phi of -K* [-4,0,0,1,1,2,1,3,1,2,4,1,1,1,1,1,0,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -6w^3z+13w^2z+15w
Inner characteristic polynomial t^6+39t^4+26t^2
Outer characteristic polynomial t^7+61t^5+103t^3
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 640*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 512*K1**2*K2**4 + 512*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 1936*K1**2*K2**2 + 1728*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K4**2 - 992*K1**2 + 960*K1*K2**3*K3 + 32*K1*K2*K3**3 + 1568*K1*K2*K3 + 152*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 752*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 608*K2**2*K3**2 - 80*K2**2*K4**2 + 328*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 364*K2**2 + 176*K2*K3*K5 + 24*K2*K4*K6 - 464*K3**2 - 146*K4**2 - 24*K5**2 - 4*K6**2 + 1000
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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