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Flat knot 6.732

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,-1,1,2,2,4,0,0,0,1,1,1,0,1,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.732']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+69t^5+119t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.732']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 640*K1**4*K2 - 1312*K1**4 + 256*K1**3*K2*K3 - 128*K1**3*K3 - 320*K1**2*K2**4 + 2080*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7248*K1**2*K2**2 - 384*K1**2*K2*K4 + 7736*K1**2*K2 - 64*K1**2*K3**2 - 4968*K1**2 + 544*K1*K2**3*K3 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5312*K1*K2*K3 + 288*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1344*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 736*K2**2*K4 - 2616*K2**2 + 88*K2*K3*K5 - 1168*K3**2 - 192*K4**2 - 8*K5**2 + 3422
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.732']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3621', 'vk6.3700', 'vk6.3893', 'vk6.4006', 'vk6.7043', 'vk6.7088', 'vk6.7265', 'vk6.7376', 'vk6.17695', 'vk6.17742', 'vk6.24242', 'vk6.24301', 'vk6.36537', 'vk6.36612', 'vk6.43643', 'vk6.43748', 'vk6.48249', 'vk6.48324', 'vk6.48409', 'vk6.48430', 'vk6.50005', 'vk6.50042', 'vk6.50127', 'vk6.50150', 'vk6.55727', 'vk6.55782', 'vk6.60299', 'vk6.60380', 'vk6.65435', 'vk6.65462', 'vk6.68563', 'vk6.68590']
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 O1O2O3O4U1O5U4U6U5O6U2U3
R3 orbit {'O1O2O3O4U1O5U4U6U5O6U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3O5U6U5U1O6U4
Gauss code of K* O1O2O3U2O4O5U6U4U5U1O6U3
Gauss code of -K* O1O2O3U1O4U3U5U6U4O5O6U2
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 2 0 2 -1],[ 3 0 2 3 1 1 3],[ 0 -2 0 1 -1 2 -1],[-2 -3 -1 0 -1 2 -3],[ 0 -1 1 1 0 1 -1],[-2 -1 -2 -2 -1 0 -2],[ 1 -3 1 3 1 2 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 2 -1 -1 -3 -3],[-2 -2 0 -1 -2 -2 -1],[ 0 1 1 0 1 -1 -1],[ 0 1 2 -1 0 -1 -2],[ 1 3 2 1 1 0 -3],[ 3 3 1 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-2,1,1,3,3,1,2,2,1,-1,1,1,1,2,3]
Phi over symmetry [-3,-1,0,0,2,2,-1,1,2,2,4,0,0,0,1,1,1,0,1,1,-2]
Phi of -K [-3,-1,0,0,2,2,-1,1,2,2,4,0,0,0,1,1,1,0,1,1,-2]
Phi of K* [-2,-2,0,0,1,3,-2,0,1,1,4,1,1,0,2,-1,0,1,0,2,-1]
Phi of -K* [-3,-1,0,0,2,2,3,1,2,1,3,1,1,2,3,1,1,1,2,1,-2]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 4z^2+23z+31
Enhanced Jones-Krushkal polynomial 4w^3z^2+23w^2z+31w
Inner characteristic polynomial t^6+51t^4+77t^2+1
Outer characteristic polynomial t^7+69t^5+119t^3+6t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -384*K1**4*K2**2 + 640*K1**4*K2 - 1312*K1**4 + 256*K1**3*K2*K3 - 128*K1**3*K3 - 320*K1**2*K2**4 + 2080*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7248*K1**2*K2**2 - 384*K1**2*K2*K4 + 7736*K1**2*K2 - 64*K1**2*K3**2 - 4968*K1**2 + 544*K1*K2**3*K3 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5312*K1*K2*K3 + 288*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1344*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 736*K2**2*K4 - 2616*K2**2 + 88*K2*K3*K5 - 1168*K3**2 - 192*K4**2 - 8*K5**2 + 3422
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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