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Flat knot 6.933

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,0,1,0,2,3,0,1,0,0,0,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.933', '7.16068']
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+91t^5+240t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.933', '7.16068']
2-strand cable arrow polynomial of the knot is: -512*K1**4 - 1536*K1**2*K2**6 + 1536*K1**2*K2**5 - 1408*K1**2*K2**4 + 256*K1**2*K2**3 - 928*K1**2*K2**2 + 1184*K1**2*K2 - 128*K1**2*K3**2 - 368*K1**2 + 1024*K1*K2**5*K3 + 256*K1*K2**3*K3 + 864*K1*K2*K3 + 96*K1*K3*K4 - 768*K2**8 + 512*K2**6*K4 - 768*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 + 512*K2**4 - 96*K2**2*K3**2 + 192*K2**2*K4 - 256*K2**2 + 64*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 :['6.933']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.467', 'vk6.525', 'vk6.920', 'vk6.1023', 'vk6.1696', 'vk6.2100', 'vk6.2209', 'vk6.2529', 'vk6.2818', 'vk6.3159', 'vk6.20290', 'vk6.21623', 'vk6.27580', 'vk6.29137', 'vk6.38998', 'vk6.41248', 'vk6.45764', 'vk6.57143', 'vk6.61767', 'vk6.66767']
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 O1O2O3O4U5U6O5O6U1U2U3U4
R3 orbit {'O1O2O3O4U5U6O5O6U1U2U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U3U4O5O6U5U6
Gauss code of K* O1O2O3O4U1U2U3U4O5O6U5U6
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 -3 -1 1 3 -1 1],[ 3 0 1 2 3 2 4],[ 1 -1 0 1 2 0 2],[-1 -2 -1 0 1 -2 0],[-3 -3 -2 -1 0 -4 -2],[ 1 -2 0 2 4 0 1],[-1 -4 -2 0 2 -1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 -1 -2 -2 -4 -3],[-1 1 0 0 -1 -2 -2],[-1 2 0 0 -2 -1 -4],[ 1 2 1 2 0 0 -1],[ 1 4 2 1 0 0 -2],[ 3 3 2 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,1,2,2,4,3,0,1,2,2,2,1,4,0,1,2]
Phi over symmetry [-3,-1,-1,1,1,3,0,1,0,2,3,0,1,0,0,0,1,2,0,0,1]
Phi of -K [-3,-1,-1,1,1,3,0,1,0,2,3,0,1,0,0,0,1,2,0,0,1]
Phi of K* [-3,-1,-1,1,1,3,0,1,0,2,3,0,1,0,0,0,1,2,0,0,1]
Phi of -K* [-3,-1,-1,1,1,3,1,2,2,4,3,0,1,2,2,2,1,4,0,1,2]
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+69t^4+180t^2
Outer characteristic polynomial t^7+91t^5+240t^3
Flat arrow polynomial -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
2-strand cable arrow polynomial -512*K1**4 - 1536*K1**2*K2**6 + 1536*K1**2*K2**5 - 1408*K1**2*K2**4 + 256*K1**2*K2**3 - 928*K1**2*K2**2 + 1184*K1**2*K2 - 128*K1**2*K3**2 - 368*K1**2 + 1024*K1*K2**5*K3 + 256*K1*K2**3*K3 + 864*K1*K2*K3 + 96*K1*K3*K4 - 768*K2**8 + 512*K2**6*K4 - 768*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 + 512*K2**4 - 96*K2**2*K3**2 + 192*K2**2*K4 - 256*K2**2 + 64*K2*K3*K5 - 168*K3**2 - 32*K4**2 - 8*K5**2 + 414
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice True
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