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Flat knot 6.178

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,2,3,4,2,0,1,1,1,1,1,2,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.178']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500']
Outer characteristic polynomial of the knot is: t^7+73t^5+84t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.178']
2-strand cable arrow polynomial of the knot is: 384*K1**4*K2 - 768*K1**4 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2752*K1**2*K2**2 - 640*K1**2*K2*K4 + 4184*K1**2*K2 - 768*K1**2*K3**2 - 64*K1**2*K3*K5 - 96*K1**2*K4**2 - 3616*K1**2 + 256*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5104*K1*K2*K3 + 1984*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**4*K4**2 + 256*K2**4*K4 - 1088*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 736*K2**2*K3**2 - 464*K2**2*K4**2 + 2040*K2**2*K4 - 8*K2**2*K6**2 - 2976*K2**2 + 472*K2*K3*K5 + 208*K2*K4*K6 - 1808*K3**2 - 1026*K4**2 - 80*K5**2 - 16*K6**2 + 3136
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.178']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17139', 'vk6.17380', 'vk6.20600', 'vk6.22012', 'vk6.23548', 'vk6.23884', 'vk6.28066', 'vk6.29521', 'vk6.35716', 'vk6.36133', 'vk6.39476', 'vk6.41679', 'vk6.43047', 'vk6.43351', 'vk6.46063', 'vk6.47728', 'vk6.55284', 'vk6.55530', 'vk6.57476', 'vk6.58640', 'vk6.59710', 'vk6.60050', 'vk6.62149', 'vk6.63109', 'vk6.65093', 'vk6.65277', 'vk6.67000', 'vk6.67865', 'vk6.68339', 'vk6.68485', 'vk6.69616', 'vk6.70308']
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 O1O2O3O4O5U1O6U5U4U6U3U2
R3 orbit {'O1O2O3O4O5U1O6U5U4U6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U3U6U2U1O6U5
Gauss code of K* O1O2O3O4O5U6U5U4U2U1O6U3
Gauss code of -K* O1O2O3O4O5U3O6U5U4U2U1U6
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 1 1 0 0 2],[ 4 0 4 3 2 1 2],[-1 -4 0 0 -1 -1 2],[-1 -3 0 0 -1 -1 2],[ 0 -2 1 1 0 0 2],[ 0 -1 1 1 0 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 0 -4],[-2 0 -2 -2 -1 -2 -2],[-1 2 0 0 -1 -1 -3],[-1 2 0 0 -1 -1 -4],[ 0 1 1 1 0 0 -1],[ 0 2 1 1 0 0 -2],[ 4 2 3 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,0,4,2,2,1,2,2,0,1,1,3,1,1,4,0,1,2]
Phi over symmetry [-4,0,0,1,1,2,1,2,3,4,2,0,1,1,1,1,1,2,0,2,2]
Phi of -K [-4,0,0,1,1,2,2,3,1,2,4,0,0,0,0,0,0,1,0,-1,-1]
Phi of K* [-2,-1,-1,0,0,4,-1,-1,0,1,4,0,0,0,1,0,0,2,0,2,3]
Phi of -K* [-4,0,0,1,1,2,1,2,3,4,2,0,1,1,1,1,1,2,0,2,2]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+51t^4+15t^2+1
Outer characteristic polynomial t^7+73t^5+84t^3+10t
Flat arrow polynomial 4*K1**2*K2 - 2*K1*K3 - K2
2-strand cable arrow polynomial 384*K1**4*K2 - 768*K1**4 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2752*K1**2*K2**2 - 640*K1**2*K2*K4 + 4184*K1**2*K2 - 768*K1**2*K3**2 - 64*K1**2*K3*K5 - 96*K1**2*K4**2 - 3616*K1**2 + 256*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5104*K1*K2*K3 + 1984*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**4*K4**2 + 256*K2**4*K4 - 1088*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 736*K2**2*K3**2 - 464*K2**2*K4**2 + 2040*K2**2*K4 - 8*K2**2*K6**2 - 2976*K2**2 + 472*K2*K3*K5 + 208*K2*K4*K6 - 1808*K3**2 - 1026*K4**2 - 80*K5**2 - 16*K6**2 + 3136
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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