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Flat knot 6.113

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,1,1,2,3,1,1,1,1,1,1,2,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.113', '7.10415']
Arrow polynomial of the knot is: -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.113', '6.132', '6.220', '6.933', '6.1250', '6.1905']
Outer characteristic polynomial of the knot is: t^7+83t^5+80t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.113']
2-strand cable arrow polynomial of the knot is: -640*K1**4 - 3584*K1**2*K2**6 + 2560*K1**2*K2**5 - 384*K1**2*K2**4 - 256*K1**2*K2**3 - 1312*K1**2*K2**2 + 1440*K1**2*K2 - 256*K1**2*K3**2 - 304*K1**2 + 2048*K1*K2**5*K3 - 256*K1*K2**3*K3 + 1184*K1*K2*K3 + 160*K1*K3*K4 - 2816*K2**8 + 1536*K2**6*K4 + 1280*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 - 256*K2**4*K4 - 256*K2**4 - 224*K2**2*K3**2 + 320*K2**2*K4 - 128*K2**2 + 128*K2*K3*K5 - 168*K3**2 - 32*K4**2 - 8*K5**2 + 414
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.113']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.468', 'vk6.526', 'vk6.923', 'vk6.1024', 'vk6.1701', 'vk6.2102', 'vk6.2210', 'vk6.2527', 'vk6.2815', 'vk6.3158', 'vk6.20291', 'vk6.21624', 'vk6.27583', 'vk6.29138', 'vk6.39001', 'vk6.41251', 'vk6.45767', 'vk6.57142', 'vk6.61764', 'vk6.66766']
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 O1O2O3O4O5O6U3U4U5U6U1U2
R3 orbit {'O1O2O3O4O5O6U3U4U5U6U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U6U1U2U3U4
Gauss code of K* O1O2O3O4O5O6U5U6U1U2U3U4
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 1 -3 -1 1 3],[ 1 0 1 -3 -1 1 3],[-1 -1 0 -3 -1 1 3],[ 3 3 3 0 1 2 3],[ 1 1 1 -1 0 1 2],[-1 -1 -1 -2 -1 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 -1 -3 -2 -3 -3],[-1 1 0 -1 -1 -1 -2],[-1 3 1 0 -1 -1 -3],[ 1 2 1 1 0 1 -1],[ 1 3 1 1 -1 0 -3],[ 3 3 2 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,1,3,2,3,3,1,1,1,2,1,1,3,-1,1,3]
Phi over symmetry [-3,-1,-1,1,1,3,-1,1,1,2,3,1,1,1,1,1,1,2,-1,-1,1]
Phi of -K [-3,-1,-1,1,1,3,-1,1,1,2,3,1,1,1,1,1,1,2,-1,-1,1]
Phi of K* [-3,-1,-1,1,1,3,-1,1,1,2,3,1,1,1,1,1,1,2,-1,-1,1]
Phi of -K* [-3,-1,-1,1,1,3,1,3,2,3,3,1,1,1,2,1,1,3,-1,1,3]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial 12w^4z-12w^3z+w^2z+3w
Inner characteristic polynomial t^6+61t^4+20t^2
Outer characteristic polynomial t^7+83t^5+80t^3
Flat arrow polynomial -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
2-strand cable arrow polynomial -640*K1**4 - 3584*K1**2*K2**6 + 2560*K1**2*K2**5 - 384*K1**2*K2**4 - 256*K1**2*K2**3 - 1312*K1**2*K2**2 + 1440*K1**2*K2 - 256*K1**2*K3**2 - 304*K1**2 + 2048*K1*K2**5*K3 - 256*K1*K2**3*K3 + 1184*K1*K2*K3 + 160*K1*K3*K4 - 2816*K2**8 + 1536*K2**6*K4 + 1280*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 - 256*K2**4*K4 - 256*K2**4 - 224*K2**2*K3**2 + 320*K2**2*K4 - 128*K2**2 + 128*K2*K3*K5 - 168*K3**2 - 32*K4**2 - 8*K5**2 + 414
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}]]
If K is slice True
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