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Flat knot 6.745

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,2,1,3,4,2,0,1,2,2,3,3,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.745']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370']
Outer characteristic polynomial of the knot is: t^7+89t^5+131t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.745']
2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 128*K1**4 - 192*K1**2*K2**4 + 1024*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5392*K1**2*K2**2 - 608*K1**2*K2*K4 + 6296*K1**2*K2 - 64*K1**2*K4**2 - 4712*K1**2 + 352*K1*K2**3*K3 - 672*K1*K2**2*K3 - 96*K1*K2**2*K5 + 4776*K1*K2*K3 + 392*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1088*K2**4 - 240*K2**2*K3**2 - 8*K2**2*K4**2 + 1048*K2**2*K4 - 2664*K2**2 + 80*K2*K3*K5 - 1000*K3**2 - 264*K4**2 + 2974
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.745']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4741', 'vk6.5069', 'vk6.6277', 'vk6.6718', 'vk6.8238', 'vk6.8687', 'vk6.9629', 'vk6.9947', 'vk6.20648', 'vk6.22079', 'vk6.28134', 'vk6.29563', 'vk6.39572', 'vk6.41803', 'vk6.46187', 'vk6.47805', 'vk6.48773', 'vk6.48983', 'vk6.49581', 'vk6.49788', 'vk6.50783', 'vk6.50996', 'vk6.51269', 'vk6.51469', 'vk6.57564', 'vk6.58734', 'vk6.62238', 'vk6.63184', 'vk6.67042', 'vk6.67915', 'vk6.69667', 'vk6.70348']
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 O1O2O3O4U2O5U1U6U5O6U4U3
R3 orbit {'O1O2O3O4U2O5U1U6U5O6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U1O5U6U5U4O6U3
Gauss code of K* O1O2O3U2O4O5U1U6U5U4O6U3
Gauss code of -K* O1O2O3U1O4U5U6U4U3O6O5U2
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 -2 2 2 2 -1],[ 3 0 0 4 3 1 2],[ 2 0 0 2 1 0 2],[-2 -4 -2 0 0 1 -3],[-2 -3 -1 0 0 1 -3],[-2 -1 0 -1 -1 0 -2],[ 1 -2 -2 3 3 2 0]]
Primitive based matrix [[ 0 2 2 2 -1 -2 -3],[-2 0 1 0 -3 -1 -3],[-2 -1 0 -1 -2 0 -1],[-2 0 1 0 -3 -2 -4],[ 1 3 2 3 0 -2 -2],[ 2 1 0 2 2 0 0],[ 3 3 1 4 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,2,3,-1,0,3,1,3,1,2,0,1,3,2,4,2,2,0]
Phi over symmetry [-3,-2,-1,2,2,2,0,2,1,3,4,2,0,1,2,2,3,3,-1,-1,0]
Phi of -K [-3,-2,-1,2,2,2,1,0,1,2,4,-1,2,3,4,0,0,1,0,-1,-1]
Phi of K* [-2,-2,-2,1,2,3,-1,-1,1,4,4,0,0,2,1,0,3,2,-1,0,1]
Phi of -K* [-3,-2,-1,2,2,2,0,2,1,3,4,2,0,1,2,2,3,3,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 8z^2+27z+23
Enhanced Jones-Krushkal polynomial 8w^3z^2-2w^3z+29w^2z+23w
Inner characteristic polynomial t^6+63t^4+79t^2+1
Outer characteristic polynomial t^7+89t^5+131t^3+10t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial 32*K1**4*K2 - 128*K1**4 - 192*K1**2*K2**4 + 1024*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5392*K1**2*K2**2 - 608*K1**2*K2*K4 + 6296*K1**2*K2 - 64*K1**2*K4**2 - 4712*K1**2 + 352*K1*K2**3*K3 - 672*K1*K2**2*K3 - 96*K1*K2**2*K5 + 4776*K1*K2*K3 + 392*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1088*K2**4 - 240*K2**2*K3**2 - 8*K2**2*K4**2 + 1048*K2**2*K4 - 2664*K2**2 + 80*K2*K3*K5 - 1000*K3**2 - 264*K4**2 + 2974
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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