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Flat knot 6.672

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,0,2,2,3,0,1,1,2,1,0,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.672', '7.31484']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.324', '6.672', '6.953', '6.1196', '6.1215', '6.1216', '6.1699']
Outer characteristic polynomial of the knot is: t^7+59t^5+46t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.601', '6.672']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 512*K1**4*K2 - 928*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 992*K1**2*K2**3 - 5104*K1**2*K2**2 - 256*K1**2*K2*K4 + 5968*K1**2*K2 - 160*K1**2*K3**2 - 112*K1**2*K4**2 - 4016*K1**2 + 416*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4600*K1*K2*K3 - 32*K1*K2*K4*K5 + 672*K1*K3*K4 + 152*K1*K4*K5 + 8*K1*K5*K6 - 96*K2**6 + 192*K2**4*K4 - 1616*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 96*K2**2*K4**2 + 1696*K2**2*K4 - 2654*K2**2 + 272*K2*K3*K5 + 72*K2*K4*K6 - 1184*K3**2 - 572*K4**2 - 88*K5**2 - 18*K6**2 + 3122
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.672']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11289', 'vk6.11369', 'vk6.12554', 'vk6.12667', 'vk6.18345', 'vk6.18683', 'vk6.24788', 'vk6.25245', 'vk6.30967', 'vk6.31094', 'vk6.32149', 'vk6.32270', 'vk6.36974', 'vk6.37429', 'vk6.44161', 'vk6.44482', 'vk6.52061', 'vk6.52142', 'vk6.52900', 'vk6.52963', 'vk6.56130', 'vk6.56355', 'vk6.60653', 'vk6.60994', 'vk6.63670', 'vk6.63717', 'vk6.64102', 'vk6.64149', 'vk6.65788', 'vk6.66045', 'vk6.68791', 'vk6.69000']
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 O1O2O3O4U2O5U6U4O6U1U5U3
R3 orbit {'O1O2O3O4U2O5U6U4O6U1U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U4O6U1U6O5U3
Gauss code of K* O1O2O3U1U4U3U5O4U2O6O5U6
Gauss code of -K* O1O2O3U4O5O4U2O6U5U1U6U3
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 2 1 2 -1],[ 2 0 -1 3 2 2 1],[ 2 1 0 2 1 1 1],[-2 -3 -2 0 0 1 -3],[-1 -2 -1 0 0 0 -1],[-2 -2 -1 -1 0 0 -2],[ 1 -1 -1 3 1 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 1 0 -3 -2 -3],[-2 -1 0 0 -2 -1 -2],[-1 0 0 0 -1 -1 -2],[ 1 3 2 1 0 -1 -1],[ 2 2 1 1 1 0 1],[ 2 3 2 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,-1,0,3,2,3,0,2,1,2,1,1,2,1,1,-1]
Phi over symmetry [-2,-2,-1,1,2,2,-1,0,2,2,3,0,1,1,2,1,0,1,1,1,-1]
Phi of -K [-2,-2,-1,1,2,2,-1,0,2,2,3,0,1,1,2,1,0,1,1,1,-1]
Phi of K* [-2,-2,-1,1,2,2,-1,1,1,2,3,1,0,1,2,1,1,2,0,0,-1]
Phi of -K* [-2,-2,-1,1,2,2,-1,1,2,2,3,1,1,1,2,1,2,3,0,0,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+41t^4+28t^2+1
Outer characteristic polynomial t^7+59t^5+46t^3+5t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -192*K1**4*K2**2 + 512*K1**4*K2 - 928*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 992*K1**2*K2**3 - 5104*K1**2*K2**2 - 256*K1**2*K2*K4 + 5968*K1**2*K2 - 160*K1**2*K3**2 - 112*K1**2*K4**2 - 4016*K1**2 + 416*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4600*K1*K2*K3 - 32*K1*K2*K4*K5 + 672*K1*K3*K4 + 152*K1*K4*K5 + 8*K1*K5*K6 - 96*K2**6 + 192*K2**4*K4 - 1616*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 96*K2**2*K4**2 + 1696*K2**2*K4 - 2654*K2**2 + 272*K2*K3*K5 + 72*K2*K4*K6 - 1184*K3**2 - 572*K4**2 - 88*K5**2 - 18*K6**2 + 3122
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice True
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