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Flat knot 6.144

Min(phi) over symmetries of the knot is: [-4,-3,-1,2,3,3,0,1,4,3,4,1,3,2,3,2,1,2,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.144']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.105', '6.144', '6.261', '6.285', '6.392', '6.480']
Outer characteristic polynomial of the knot is: t^7+126t^5+76t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.144']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 704*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 + 896*K1**2*K2**3 - 3808*K1**2*K2**2 - 864*K1**2*K2*K4 + 3632*K1**2*K2 - 192*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 2384*K1**2 + 288*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 800*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 3840*K1*K2*K3 - 64*K1*K2*K4*K5 + 960*K1*K3*K4 + 216*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1168*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 416*K2**2*K3**2 - 440*K2**2*K4**2 + 1552*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K6**2 - 1622*K2**2 + 520*K2*K3*K5 + 152*K2*K4*K6 + 16*K2*K5*K7 - 944*K3**2 - 570*K4**2 - 136*K5**2 - 10*K6**2 + 1952
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.144']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73976', 'vk6.73989', 'vk6.74491', 'vk6.74512', 'vk6.75947', 'vk6.75974', 'vk6.76707', 'vk6.76727', 'vk6.78947', 'vk6.78962', 'vk6.79493', 'vk6.79510', 'vk6.80474', 'vk6.80488', 'vk6.80951', 'vk6.80962', 'vk6.83014', 'vk6.83100', 'vk6.83655', 'vk6.83784', 'vk6.83946', 'vk6.84122', 'vk6.84269', 'vk6.85175', 'vk6.85542', 'vk6.85868', 'vk6.86256', 'vk6.86583', 'vk6.86742', 'vk6.87442', 'vk6.88306', 'vk6.89745']
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 O1O2O3O4O5U1O6U2U3U6U5U4
R3 orbit {'O1O2O3O4O5U1O6U2U3U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U1U6U3U4O6U5
Gauss code of K* O1O2O3O4O5U6U1U2U5U4O6U3
Gauss code of -K* O1O2O3O4O5U3O6U2U1U4U5U6
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 -4 -3 -1 3 3 2],[ 4 0 1 2 4 3 2],[ 3 -1 0 1 4 3 2],[ 1 -2 -1 0 3 2 1],[-3 -4 -4 -3 0 0 0],[-3 -3 -3 -2 0 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 3 2 -1 -3 -4],[-3 0 0 0 -2 -3 -3],[-3 0 0 0 -3 -4 -4],[-2 0 0 0 -1 -2 -2],[ 1 2 3 1 0 -1 -2],[ 3 3 4 2 1 0 -1],[ 4 3 4 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-2,1,3,4,0,0,2,3,3,0,3,4,4,1,2,2,1,2,1]
Phi over symmetry [-4,-3,-1,2,3,3,0,1,4,3,4,1,3,2,3,2,1,2,1,1,0]
Phi of -K [-4,-3,-1,2,3,3,0,1,4,3,4,1,3,2,3,2,1,2,1,1,0]
Phi of K* [-3,-3,-2,1,3,4,0,1,1,2,3,1,2,3,4,2,3,4,1,1,0]
Phi of -K* [-4,-3,-1,2,3,3,1,2,2,3,4,1,2,3,4,1,2,3,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+78t^4+15t^2
Outer characteristic polynomial t^7+126t^5+76t^3+4t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
2-strand cable arrow polynomial -256*K1**4*K2**2 + 480*K1**4*K2 - 704*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 + 896*K1**2*K2**3 - 3808*K1**2*K2**2 - 864*K1**2*K2*K4 + 3632*K1**2*K2 - 192*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 2384*K1**2 + 288*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 800*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 3840*K1*K2*K3 - 64*K1*K2*K4*K5 + 960*K1*K3*K4 + 216*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1168*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 416*K2**2*K3**2 - 440*K2**2*K4**2 + 1552*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K6**2 - 1622*K2**2 + 520*K2*K3*K5 + 152*K2*K4*K6 + 16*K2*K5*K7 - 944*K3**2 - 570*K4**2 - 136*K5**2 - 10*K6**2 + 1952
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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