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Flat knot 6.409

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,0,-1,1,1,2,-1,1,2,3,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.409', '7.19809']
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+43t^5+85t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.409', '7.19809']
2-strand cable arrow polynomial of the knot is: -640*K1**2*K2**4 + 960*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 5856*K1**2*K2**2 - 384*K1**2*K2*K4 + 4352*K1**2*K2 - 256*K1**2*K4**2 - 2128*K1**2 + 960*K1*K2**3*K3 - 960*K1*K2**2*K3 - 512*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4160*K1*K2*K3 + 496*K1*K3*K4 + 240*K1*K4*K5 - 960*K2**6 + 1696*K2**4*K4 - 5216*K2**4 - 448*K2**3*K6 - 160*K2**2*K3**2 - 656*K2**2*K4**2 + 3880*K2**2*K4 + 288*K2**2 + 128*K2*K3*K5 + 280*K2*K4*K6 - 624*K3**2 - 600*K4**2 - 32*K5**2 - 16*K6**2 + 1750
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.409']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.325', 'vk6.362', 'vk6.367', 'vk6.719', 'vk6.764', 'vk6.773', 'vk6.1453', 'vk6.1510', 'vk6.1517', 'vk6.1955', 'vk6.1992', 'vk6.1997', 'vk6.2460', 'vk6.2662', 'vk6.2998', 'vk6.3122', 'vk6.18395', 'vk6.18398', 'vk6.18733', 'vk6.18738', 'vk6.24854', 'vk6.24859', 'vk6.25313', 'vk6.25322', 'vk6.37050', 'vk6.37055', 'vk6.44205', 'vk6.44208', 'vk6.56169', 'vk6.56172', 'vk6.60703', 'vk6.60706']
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 O1O2O3O4O5U4U5O6U2U1U6U3
R3 orbit {'O1O2O3O4O5U4U5O6U2U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U5U4O6U1U2
Gauss code of K* O1O2O3O4U2U1U4U5U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U6U1U4U3
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 -2 -2 2 -1 1 2],[ 2 0 0 3 -1 1 2],[ 2 0 0 2 -1 1 1],[-2 -3 -2 0 -1 1 0],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 0 1 -1 -2 -3],[-2 0 0 0 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[ 1 1 0 1 0 1 1],[ 2 2 1 1 -1 0 0],[ 2 3 2 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,0,-1,1,2,3,0,0,1,2,1,1,1,-1,-1,0]
Phi over symmetry [-2,-2,-1,1,2,2,0,-1,1,1,2,-1,1,2,3,1,0,1,0,-1,0]
Phi of -K [-2,-2,-1,1,2,2,0,2,2,1,2,2,2,2,3,1,2,3,2,1,0]
Phi of K* [-2,-2,-1,1,2,2,0,1,3,2,3,2,2,1,2,1,2,2,2,2,0]
Phi of -K* [-2,-2,-1,1,2,2,0,-1,1,1,2,-1,1,2,3,1,0,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+8z+5
Enhanced Jones-Krushkal polynomial -6w^4z^2+9w^3z^2-16w^3z+24w^2z+5w
Inner characteristic polynomial t^6+25t^4+33t^2+1
Outer characteristic polynomial t^7+43t^5+85t^3+19t
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -640*K1**2*K2**4 + 960*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 5856*K1**2*K2**2 - 384*K1**2*K2*K4 + 4352*K1**2*K2 - 256*K1**2*K4**2 - 2128*K1**2 + 960*K1*K2**3*K3 - 960*K1*K2**2*K3 - 512*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4160*K1*K2*K3 + 496*K1*K3*K4 + 240*K1*K4*K5 - 960*K2**6 + 1696*K2**4*K4 - 5216*K2**4 - 448*K2**3*K6 - 160*K2**2*K3**2 - 656*K2**2*K4**2 + 3880*K2**2*K4 + 288*K2**2 + 128*K2*K3*K5 + 280*K2*K4*K6 - 624*K3**2 - 600*K4**2 - 32*K5**2 - 16*K6**2 + 1750
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
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