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Flat knot 6.257

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,1,2,3,2,4,1,2,1,2,0,0,-1,0,0,-2]
Flat knots (up to 7 crossings) with same phi are :['6.257']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**2*K2 + 6*K1**2 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.221', '6.257']
Outer characteristic polynomial of the knot is: t^7+93t^5+107t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.257']
2-strand cable arrow polynomial of the knot is: -64*K1**4 - 1408*K1**2*K2**4 + 2912*K1**2*K2**3 - 4672*K1**2*K2**2 - 352*K1**2*K2*K4 + 3392*K1**2*K2 - 192*K1**2*K3**2 - 2320*K1**2 + 1440*K1*K2**3*K3 - 1440*K1*K2**2*K3 - 224*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3360*K1*K2*K3 + 600*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1088*K2**6 - 32*K2**4*K4**2 + 1120*K2**4*K4 - 2504*K2**4 - 128*K2**3*K6 - 384*K2**2*K3**2 - 304*K2**2*K4**2 + 2032*K2**2*K4 - 480*K2**2 + 192*K2*K3*K5 + 72*K2*K4*K6 - 784*K3**2 - 450*K4**2 - 48*K5**2 + 1760
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.257']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73193', 'vk6.73207', 'vk6.73625', 'vk6.74310', 'vk6.74407', 'vk6.74954', 'vk6.75018', 'vk6.75107', 'vk6.75124', 'vk6.75559', 'vk6.75585', 'vk6.76520', 'vk6.76591', 'vk6.76931', 'vk6.78045', 'vk6.78063', 'vk6.78529', 'vk6.78555', 'vk6.79358', 'vk6.79780', 'vk6.79854', 'vk6.79963', 'vk6.80814', 'vk6.80885', 'vk6.83684', 'vk6.84710', 'vk6.84816', 'vk6.85266', 'vk6.85636', 'vk6.87701', 'vk6.88383', 'vk6.89485']
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 O1O2O3O4O5U1U2O6U5U4U6U3
R3 orbit {'O1O2O3O4O5U1U2O6U5U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U2U1O6U4U5
Gauss code of K* O1O2O3O4U5U6U4U2U1O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U4U3U1U5U6
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 -4 -2 2 1 1 2],[ 4 0 1 4 3 2 2],[ 2 -1 0 3 2 1 2],[-2 -4 -3 0 -1 -1 2],[-1 -3 -2 1 0 0 2],[-1 -2 -1 1 0 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 2 -1 -1 -3 -4],[-2 -2 0 -1 -2 -2 -2],[-1 1 1 0 0 -1 -2],[-1 1 2 0 0 -2 -3],[ 2 3 2 1 2 0 -1],[ 4 4 2 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,-2,1,1,3,4,1,2,2,2,0,1,2,2,3,1]
Phi over symmetry [-4,-2,1,1,2,2,1,2,3,2,4,1,2,1,2,0,0,-1,0,0,-2]
Phi of -K [-4,-2,1,1,2,2,1,2,3,2,4,1,2,1,2,0,0,-1,0,0,-2]
Phi of K* [-2,-2,-1,-1,2,4,-2,-1,0,2,4,0,0,1,2,0,1,2,2,3,1]
Phi of -K* [-4,-2,1,1,2,2,1,2,3,2,4,1,2,2,3,0,1,1,2,1,-2]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial -4w^4z^2+10w^3z^2-4w^3z+23w^2z+15w
Inner characteristic polynomial t^6+63t^4+12t^2
Outer characteristic polynomial t^7+93t^5+107t^3+7t
Flat arrow polynomial -8*K1**4 + 4*K1**2*K2 + 6*K1**2 - K2
2-strand cable arrow polynomial -64*K1**4 - 1408*K1**2*K2**4 + 2912*K1**2*K2**3 - 4672*K1**2*K2**2 - 352*K1**2*K2*K4 + 3392*K1**2*K2 - 192*K1**2*K3**2 - 2320*K1**2 + 1440*K1*K2**3*K3 - 1440*K1*K2**2*K3 - 224*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3360*K1*K2*K3 + 600*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1088*K2**6 - 32*K2**4*K4**2 + 1120*K2**4*K4 - 2504*K2**4 - 128*K2**3*K6 - 384*K2**2*K3**2 - 304*K2**2*K4**2 + 2032*K2**2*K4 - 480*K2**2 + 192*K2*K3*K5 + 72*K2*K4*K6 - 784*K3**2 - 450*K4**2 - 48*K5**2 + 1760
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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