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Flat knot 6.834

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,0,1,1,2,4,1,1,0,1,0,0,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.834']
Arrow polynomial of the knot is: 16*K1**3 - 4*K1**2 - 8*K1*K2 - 8*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.834']
Outer characteristic polynomial of the knot is: t^7+77t^5+95t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.834']
2-strand cable arrow polynomial of the knot is: -128*K1**4 - 1280*K1**2*K2**4 + 3072*K1**2*K2**3 - 7072*K1**2*K2**2 - 192*K1**2*K2*K4 + 4912*K1**2*K2 - 2816*K1**2 + 2304*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 384*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 4192*K1*K2*K3 + 48*K1*K3*K4 + 16*K1*K4*K5 - 3584*K2**6 + 3328*K2**4*K4 - 5680*K2**4 - 448*K2**3*K6 - 672*K2**2*K3**2 - 352*K2**2*K4**2 + 3728*K2**2*K4 + 1248*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 664*K3**2 - 292*K4**2 - 8*K5**2 + 2034
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.834']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81094', 'vk6.81096', 'vk6.81124', 'vk6.81128', 'vk6.81201', 'vk6.81204', 'vk6.81247', 'vk6.82055', 'vk6.82553', 'vk6.83020', 'vk6.83491', 'vk6.83493', 'vk6.83955', 'vk6.83995', 'vk6.86308', 'vk6.86310', 'vk6.88524', 'vk6.88848', 'vk6.88851', 'vk6.89890']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U1U2O5O6U3U4U5U6
R3 orbit {'O1O2O3O4U1U2O5O6U3U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1U2O5O6U3U4
Gauss code of K* O1O2O3O4U5U6U1U2O5O6U3U4
Gauss code of -K* Same
Diagrammatic symmetry type -
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 -1 -1 1 1 3],[ 3 0 1 2 3 2 2],[ 1 -1 0 1 2 2 2],[ 1 -2 -1 0 1 2 3],[-1 -3 -2 -1 0 1 2],[-1 -2 -2 -2 -1 0 1],[-3 -2 -2 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 -1 -2 -2 -3 -2],[-1 1 0 -1 -2 -2 -2],[-1 2 1 0 -2 -1 -3],[ 1 2 2 2 0 1 -1],[ 1 3 2 1 -1 0 -2],[ 3 2 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,1,2,2,3,2,1,2,2,2,2,1,3,-1,1,2]
Phi over symmetry [-3,-1,-1,1,1,3,0,1,1,2,4,1,1,0,1,0,0,2,-1,0,1]
Phi of -K [-3,-1,-1,1,1,3,0,1,1,2,4,1,1,0,1,0,0,2,-1,0,1]
Phi of K* [-3,-1,-1,1,1,3,0,1,1,2,4,1,1,0,1,0,0,2,-1,0,1]
Phi of -K* [-3,-1,-1,1,1,3,1,2,2,3,2,1,2,2,2,2,1,3,-1,1,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -6w^4z^2+7w^3z^2-16w^3z+22w^2z+9w
Inner characteristic polynomial t^6+55t^4+23t^2+1
Outer characteristic polynomial t^7+77t^5+95t^3+19t
Flat arrow polynomial 16*K1**3 - 4*K1**2 - 8*K1*K2 - 8*K1 + 2*K2 + 3
2-strand cable arrow polynomial -128*K1**4 - 1280*K1**2*K2**4 + 3072*K1**2*K2**3 - 7072*K1**2*K2**2 - 192*K1**2*K2*K4 + 4912*K1**2*K2 - 2816*K1**2 + 2304*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 384*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 4192*K1*K2*K3 + 48*K1*K3*K4 + 16*K1*K4*K5 - 3584*K2**6 + 3328*K2**4*K4 - 5680*K2**4 - 448*K2**3*K6 - 672*K2**2*K3**2 - 352*K2**2*K4**2 + 3728*K2**2*K4 + 1248*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 664*K3**2 - 292*K4**2 - 8*K5**2 + 2034
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice True
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