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Flat knot 6.768

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,1,3,1,0,0,1,0,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.768']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 6*K1*K2 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.670', '6.768']
Outer characteristic polynomial of the knot is: t^7+40t^5+40t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.768']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 320*K1**4*K2**2 + 1856*K1**4*K2 - 4960*K1**4 + 896*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 192*K1**2*K2**4 + 736*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5328*K1**2*K2**2 - 928*K1**2*K2*K4 + 9784*K1**2*K2 - 896*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 4608*K1**2 + 256*K1*K2**3*K3 - 896*K1*K2**2*K3 - 288*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6608*K1*K2*K3 + 1592*K1*K3*K4 + 256*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 472*K2**4 - 32*K2**3*K6 - 112*K2**2*K3**2 - 24*K2**2*K4**2 + 1072*K2**2*K4 - 4332*K2**2 + 328*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 1992*K3**2 - 722*K4**2 - 160*K5**2 - 20*K6**2 + 4456
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.768']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4455', 'vk6.4552', 'vk6.5841', 'vk6.5970', 'vk6.7897', 'vk6.8015', 'vk6.9328', 'vk6.9449', 'vk6.13403', 'vk6.13498', 'vk6.13691', 'vk6.14061', 'vk6.15032', 'vk6.15154', 'vk6.17800', 'vk6.17833', 'vk6.18846', 'vk6.19412', 'vk6.19707', 'vk6.24347', 'vk6.25439', 'vk6.25472', 'vk6.26590', 'vk6.33249', 'vk6.33308', 'vk6.37573', 'vk6.44873', 'vk6.48660', 'vk6.50556', 'vk6.53657', 'vk6.55817', 'vk6.65481']
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 O1O2O3O4U3O5U1U5U4O6U2U6
R3 orbit {'O1O2O3O4U3O5U1U5U4O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3O5U1U6U4O6U2
Gauss code of K* O1O2O3U4O5O4U1U5U6U3O6U2
Gauss code of -K* O1O2O3U2O4U1U4U5U3O6O5U6
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 0 -1 2 1 1],[ 3 0 3 0 3 1 1],[ 0 -3 0 -1 1 0 1],[ 1 0 1 0 1 0 0],[-2 -3 -1 -1 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -1 -1 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 0 -1],[ 0 1 0 1 0 -1 -3],[ 1 1 0 0 1 0 0],[ 3 3 1 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,0,1,1,3,0,0,0,1,1,0,1,1,3,0]
Phi over symmetry [-3,-1,0,1,1,2,0,3,1,1,3,1,0,0,1,0,1,1,0,0,0]
Phi of -K [-3,-1,0,1,1,2,2,0,3,3,2,0,2,2,2,0,1,1,0,1,1]
Phi of K* [-2,-1,-1,0,1,3,1,1,1,2,2,0,0,2,3,1,2,3,0,0,2]
Phi of -K* [-3,-1,0,1,1,2,0,3,1,1,3,1,0,0,1,0,1,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+21z+35
Enhanced Jones-Krushkal polynomial 2w^3z^2+21w^2z+35w
Inner characteristic polynomial t^6+24t^4+11t^2+1
Outer characteristic polynomial t^7+40t^5+40t^3+6t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 6*K1*K2 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -384*K1**6 - 320*K1**4*K2**2 + 1856*K1**4*K2 - 4960*K1**4 + 896*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 192*K1**2*K2**4 + 736*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5328*K1**2*K2**2 - 928*K1**2*K2*K4 + 9784*K1**2*K2 - 896*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 4608*K1**2 + 256*K1*K2**3*K3 - 896*K1*K2**2*K3 - 288*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6608*K1*K2*K3 + 1592*K1*K3*K4 + 256*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 472*K2**4 - 32*K2**3*K6 - 112*K2**2*K3**2 - 24*K2**2*K4**2 + 1072*K2**2*K4 - 4332*K2**2 + 328*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 1992*K3**2 - 722*K4**2 - 160*K5**2 - 20*K6**2 + 4456
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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