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Flat knot 6.424

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,2,1,3,4,1,1,1,1,0,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.424']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1*K2 - 4*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.134', '6.409', '6.424', '6.534', '6.942', '6.969', '6.1192', '6.1280', '6.1310', '6.1325', '6.1858', '6.1925']
Outer characteristic polynomial of the knot is: t^7+71t^5+122t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.424']
2-strand cable arrow polynomial of the knot is: 896*K1**2*K2**5 - 4480*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3840*K1**2*K2**3 - 4928*K1**2*K2**2 - 96*K1**2*K2*K4 + 3000*K1**2*K2 - 1824*K1**2 - 512*K1*K2**4*K3 + 2880*K1*K2**3*K3 - 352*K1*K2**2*K3 + 2408*K1*K2*K3 + 40*K1*K3*K4 - 1216*K2**6 + 864*K2**4*K4 - 1408*K2**4 - 352*K2**2*K3**2 - 80*K2**2*K4**2 + 648*K2**2*K4 + 104*K2**2 - 392*K3**2 - 104*K4**2 + 1190
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.424']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20132', 'vk6.20134', 'vk6.20136', 'vk6.20138', 'vk6.21426', 'vk6.21428', 'vk6.27236', 'vk6.27238', 'vk6.27244', 'vk6.27246', 'vk6.28902', 'vk6.28904', 'vk6.38648', 'vk6.38654', 'vk6.38662', 'vk6.38668', 'vk6.40863', 'vk6.40869', 'vk6.47253', 'vk6.47255', 'vk6.56955', 'vk6.56957', 'vk6.56963', 'vk6.56965', 'vk6.58113', 'vk6.58115', 'vk6.62651', 'vk6.62657', 'vk6.67452', 'vk6.67454', 'vk6.70026', 'vk6.70028']
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 O1O2O3O4O5U6U1O6U3U4U5U2
R3 orbit {'O1O2O3O4O5U6U1O6U3U4U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U1U2U3O6U5U6
Gauss code of K* O1O2O3O4U5U4U1U2U3O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U2U3U4U1U6
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 -3 1 -1 1 3 -1],[ 3 0 3 0 1 2 3],[-1 -3 0 -2 0 2 -1],[ 1 0 2 0 1 2 1],[-1 -1 0 -1 0 1 -1],[-3 -2 -2 -2 -1 0 -3],[ 1 -3 1 -1 1 3 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 -1 -2 -2 -3 -2],[-1 1 0 0 -1 -1 -1],[-1 2 0 0 -2 -1 -3],[ 1 2 1 2 0 1 0],[ 1 3 1 1 -1 0 -3],[ 3 2 1 3 0 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,1,2,2,3,2,0,1,1,1,2,1,3,-1,0,3]
Phi over symmetry [-3,-1,-1,1,1,3,-1,2,1,3,4,1,1,1,1,0,1,2,0,0,1]
Phi of -K [-3,-1,-1,1,1,3,-1,2,1,3,4,1,1,1,1,0,1,2,0,0,1]
Phi of K* [-3,-1,-1,1,1,3,0,1,1,2,4,0,1,0,1,1,1,3,-1,-1,2]
Phi of -K* [-3,-1,-1,1,1,3,0,3,1,3,2,1,1,2,2,1,1,3,0,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -6w^4z^2+8w^3z^2-10w^3z+17w^2z+7w
Inner characteristic polynomial t^6+49t^4+50t^2
Outer characteristic polynomial t^7+71t^5+122t^3+6t
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial 896*K1**2*K2**5 - 4480*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3840*K1**2*K2**3 - 4928*K1**2*K2**2 - 96*K1**2*K2*K4 + 3000*K1**2*K2 - 1824*K1**2 - 512*K1*K2**4*K3 + 2880*K1*K2**3*K3 - 352*K1*K2**2*K3 + 2408*K1*K2*K3 + 40*K1*K3*K4 - 1216*K2**6 + 864*K2**4*K4 - 1408*K2**4 - 352*K2**2*K3**2 - 80*K2**2*K4**2 + 648*K2**2*K4 + 104*K2**2 - 392*K3**2 - 104*K4**2 + 1190
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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