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Flat knot 6.577

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,4,2,3,1,2,1,2,0,1,1,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.577']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.240', '6.577', '6.625', '6.1020']
Outer characteristic polynomial of the knot is: t^7+57t^5+46t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.577']
2-strand cable arrow polynomial of the knot is: 352*K1**4*K2 - 2128*K1**4 + 608*K1**3*K2*K3 + 32*K1**3*K3*K4 - 2272*K1**3*K3 - 1584*K1**2*K2**2 - 512*K1**2*K2*K4 + 6464*K1**2*K2 - 1024*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 4892*K1**2 + 128*K1*K2**3*K3 - 96*K1*K2**2*K3 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 5768*K1*K2*K3 - 32*K1*K3**2*K5 + 1424*K1*K3*K4 + 200*K1*K4*K5 + 8*K1*K5*K6 - 72*K2**4 - 192*K2**2*K3**2 - 24*K2**2*K4**2 + 464*K2**2*K4 - 3406*K2**2 + 256*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2136*K3**2 - 638*K4**2 - 132*K5**2 - 18*K6**2 + 3652
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.577']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4640', 'vk6.4913', 'vk6.6072', 'vk6.6575', 'vk6.8101', 'vk6.8491', 'vk6.9479', 'vk6.9848', 'vk6.20633', 'vk6.22062', 'vk6.28119', 'vk6.29562', 'vk6.39543', 'vk6.41768', 'vk6.46154', 'vk6.47798', 'vk6.48680', 'vk6.48873', 'vk6.49422', 'vk6.49653', 'vk6.50694', 'vk6.50885', 'vk6.51173', 'vk6.51384', 'vk6.57525', 'vk6.58715', 'vk6.62221', 'vk6.63169', 'vk6.67027', 'vk6.67902', 'vk6.69656', 'vk6.70339']
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 O1O2O3O4U1O5U2O6U5U6U4U3
R3 orbit {'O1O2O3O4U1O5U2O6U5U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U1U5U6O5U3O6U4
Gauss code of K* O1O2O3O4U5U6U4U3O5U1O6U2
Gauss code of -K* O1O2O3O4U3O5U4O6U2U1U5U6
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 -3 -2 2 2 0 1],[ 3 0 1 3 2 1 0],[ 2 -1 0 3 2 1 1],[-2 -3 -3 0 0 -1 1],[-2 -2 -2 0 0 -1 1],[ 0 -1 -1 1 1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 0 1 -1 -2 -2],[-2 0 0 1 -1 -3 -3],[-1 -1 -1 0 -1 -1 0],[ 0 1 1 1 0 -1 -1],[ 2 2 3 1 1 0 -1],[ 3 2 3 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,0,-1,1,2,2,-1,1,3,3,1,1,0,1,1,1]
Phi over symmetry [-3,-2,0,1,2,2,0,2,4,2,3,1,2,1,2,0,1,1,2,2,0]
Phi of -K [-3,-2,0,1,2,2,0,2,4,2,3,1,2,1,2,0,1,1,2,2,0]
Phi of K* [-2,-2,-1,0,2,3,0,2,1,1,2,2,1,2,3,0,2,4,1,2,0]
Phi of -K* [-3,-2,0,1,2,2,1,1,0,2,3,1,1,2,3,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+35t^4+17t^2+1
Outer characteristic polynomial t^7+57t^5+46t^3+6t
Flat arrow polynomial -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 352*K1**4*K2 - 2128*K1**4 + 608*K1**3*K2*K3 + 32*K1**3*K3*K4 - 2272*K1**3*K3 - 1584*K1**2*K2**2 - 512*K1**2*K2*K4 + 6464*K1**2*K2 - 1024*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 4892*K1**2 + 128*K1*K2**3*K3 - 96*K1*K2**2*K3 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 5768*K1*K2*K3 - 32*K1*K3**2*K5 + 1424*K1*K3*K4 + 200*K1*K4*K5 + 8*K1*K5*K6 - 72*K2**4 - 192*K2**2*K3**2 - 24*K2**2*K4**2 + 464*K2**2*K4 - 3406*K2**2 + 256*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2136*K3**2 - 638*K4**2 - 132*K5**2 - 18*K6**2 + 3652
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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