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Flat knot 6.528

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,3,4,1,0,1,1,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.528']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.311', '6.528', '6.536', '6.817', '6.982', '6.984', '6.1284']
Outer characteristic polynomial of the knot is: t^7+52t^5+44t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.528']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 1376*K1**4*K2 - 4736*K1**4 + 384*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1312*K1**3*K3 - 192*K1**2*K2**4 + 672*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 3920*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1248*K1**2*K2*K4 + 8520*K1**2*K2 - 1472*K1**2*K3**2 - 224*K1**2*K3*K5 - 400*K1**2*K4**2 - 32*K1**2*K5**2 - 4512*K1**2 + 128*K1*K2**3*K3 - 480*K1*K2**2*K3 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6440*K1*K2*K3 + 2560*K1*K3*K4 + 592*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 376*K2**4 - 144*K2**2*K3**2 - 24*K2**2*K4**2 + 896*K2**2*K4 - 3876*K2**2 + 336*K2*K3*K5 + 48*K2*K4*K6 - 2132*K3**2 - 990*K4**2 - 252*K5**2 - 28*K6**2 + 4348
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.528']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4838', 'vk6.5181', 'vk6.6404', 'vk6.6835', 'vk6.8365', 'vk6.8793', 'vk6.9731', 'vk6.10034', 'vk6.11624', 'vk6.11977', 'vk6.12966', 'vk6.20463', 'vk6.20742', 'vk6.21817', 'vk6.27848', 'vk6.29357', 'vk6.31427', 'vk6.32601', 'vk6.39275', 'vk6.39782', 'vk6.41453', 'vk6.46342', 'vk6.47578', 'vk6.47917', 'vk6.49063', 'vk6.49893', 'vk6.51317', 'vk6.51534', 'vk6.53215', 'vk6.57322', 'vk6.62009', 'vk6.64296']
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 O1O2O3O4O5U2U5U3O6U4U1U6
R3 orbit {'O1O2O3O4O5U2U5U3O6U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U2O6U3U1U4
Gauss code of K* O1O2O3U4O5O6O4U6U1U3U5U2
Gauss code of -K* O1O2O3U1O4O5O6U5U3U4U6U2
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 -1 -3 0 1 1 2],[ 1 0 -3 0 2 1 2],[ 3 3 0 2 3 1 1],[ 0 0 -2 0 1 0 1],[-1 -2 -3 -1 0 0 1],[-1 -1 -1 0 0 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 -2 -1],[-1 0 0 0 0 -1 -1],[-1 1 0 0 -1 -2 -3],[ 0 1 0 1 0 0 -2],[ 1 2 1 2 0 0 -3],[ 3 1 1 3 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,2,1,0,0,1,1,1,2,3,0,2,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,3,4,1,0,1,1,0,1,1,0,0,1]
Phi of -K [-3,-1,0,1,1,2,-1,1,1,3,4,1,0,1,1,0,1,1,0,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,1,4,0,0,0,1,1,1,3,1,1,-1]
Phi of -K* [-3,-1,0,1,1,2,3,2,1,3,1,0,1,2,2,0,1,1,0,0,1]
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+36t^4+15t^2
Outer characteristic polynomial t^7+52t^5+44t^3+5t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -128*K1**6 + 1376*K1**4*K2 - 4736*K1**4 + 384*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1312*K1**3*K3 - 192*K1**2*K2**4 + 672*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 3920*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1248*K1**2*K2*K4 + 8520*K1**2*K2 - 1472*K1**2*K3**2 - 224*K1**2*K3*K5 - 400*K1**2*K4**2 - 32*K1**2*K5**2 - 4512*K1**2 + 128*K1*K2**3*K3 - 480*K1*K2**2*K3 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6440*K1*K2*K3 + 2560*K1*K3*K4 + 592*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 376*K2**4 - 144*K2**2*K3**2 - 24*K2**2*K4**2 + 896*K2**2*K4 - 3876*K2**2 + 336*K2*K3*K5 + 48*K2*K4*K6 - 2132*K3**2 - 990*K4**2 - 252*K5**2 - 28*K6**2 + 4348
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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