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Flat knot 6.762

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,0,3,2,4,0,1,0,1,2,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.762']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 6*K1*K2 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.388', '6.687', '6.762']
Outer characteristic polynomial of the knot is: t^7+64t^5+52t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.762']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 960*K1**4*K2**2 + 1632*K1**4*K2 - 2672*K1**4 + 288*K1**3*K2*K3 + 96*K1**3*K3*K4 - 448*K1**2*K2**4 + 1216*K1**2*K2**3 - 4320*K1**2*K2**2 + 4920*K1**2*K2 - 528*K1**2*K3**2 - 192*K1**2*K4**2 - 32*K1**2*K5**2 - 2688*K1**2 + 256*K1*K2**3*K3 + 3224*K1*K2*K3 + 952*K1*K3*K4 + 216*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 784*K2**4 - 160*K2**2*K3**2 - 24*K2**2*K4**2 + 328*K2**2*K4 - 1828*K2**2 + 216*K2*K3*K5 + 40*K2*K4*K6 - 1164*K3**2 - 500*K4**2 - 164*K5**2 - 36*K6**2 + 2802
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.762']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4437', 'vk6.4534', 'vk6.5819', 'vk6.5948', 'vk6.7878', 'vk6.7989', 'vk6.9298', 'vk6.9419', 'vk6.10158', 'vk6.10231', 'vk6.10372', 'vk6.17891', 'vk6.17954', 'vk6.18274', 'vk6.18609', 'vk6.24394', 'vk6.25160', 'vk6.30041', 'vk6.30104', 'vk6.36884', 'vk6.37342', 'vk6.43821', 'vk6.44101', 'vk6.44424', 'vk6.48626', 'vk6.50525', 'vk6.50610', 'vk6.51128', 'vk6.51669', 'vk6.55851', 'vk6.56077', 'vk6.65512']
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 O1O2O3O4U2O5U6U5U3O6U1U4
R3 orbit {'O1O2O3O4U2O5U6U5U3O6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4O5U2U6U5O6U3
Gauss code of K* O1O2O3U1O4O5U4U6U3U5O6U2
Gauss code of -K* O1O2O3U2O4U5U1U4U6O5O6U3
Diagrammatic symmetry type c
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 -2 1 3 1 -2],[ 1 0 -1 2 3 1 -1],[ 2 1 0 1 2 0 1],[-1 -2 -1 0 0 0 -2],[-3 -3 -2 0 0 1 -4],[-1 -1 0 0 -1 0 -1],[ 2 1 -1 2 4 1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 1 0 -3 -2 -4],[-1 -1 0 0 -1 0 -1],[-1 0 0 0 -2 -1 -2],[ 1 3 1 2 0 -1 -1],[ 2 2 0 1 1 0 1],[ 2 4 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,-1,0,3,2,4,0,1,0,1,2,1,2,1,1,-1]
Phi over symmetry [-3,-1,-1,1,2,2,-1,0,3,2,4,0,1,0,1,2,1,2,1,1,-1]
Phi of -K [-2,-2,-1,1,1,3,-1,0,2,3,3,0,1,2,1,0,1,1,0,2,3]
Phi of K* [-3,-1,-1,1,2,2,2,3,1,1,3,0,0,1,2,1,2,3,0,0,-1]
Phi of -K* [-2,-2,-1,1,1,3,-1,1,1,2,4,1,0,1,2,1,2,3,0,-1,0]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial -4w^3z+17w^2z+27w
Inner characteristic polynomial t^6+44t^4+20t^2
Outer characteristic polynomial t^7+64t^5+52t^3
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 6*K1*K2 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial 256*K1**4*K2**3 - 960*K1**4*K2**2 + 1632*K1**4*K2 - 2672*K1**4 + 288*K1**3*K2*K3 + 96*K1**3*K3*K4 - 448*K1**2*K2**4 + 1216*K1**2*K2**3 - 4320*K1**2*K2**2 + 4920*K1**2*K2 - 528*K1**2*K3**2 - 192*K1**2*K4**2 - 32*K1**2*K5**2 - 2688*K1**2 + 256*K1*K2**3*K3 + 3224*K1*K2*K3 + 952*K1*K3*K4 + 216*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 784*K2**4 - 160*K2**2*K3**2 - 24*K2**2*K4**2 + 328*K2**2*K4 - 1828*K2**2 + 216*K2*K3*K5 + 40*K2*K4*K6 - 1164*K3**2 - 500*K4**2 - 164*K5**2 - 36*K6**2 + 2802
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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