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Flat knot 6.269

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,1,2,4,3,1,1,2,2,-1,-1,-1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.269']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.96', '6.149', '6.269', '6.441', '6.457']
Outer characteristic polynomial of the knot is: t^7+70t^5+78t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.269']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 672*K1**4*K2 - 1104*K1**4 + 320*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**4 + 480*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 2272*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 3576*K1**2*K2 - 176*K1**2*K3**2 - 32*K1**2*K4**2 - 2236*K1**2 + 576*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 2872*K1*K2*K3 + 536*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 608*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 544*K2**2*K3**2 - 32*K2**2*K3*K7 - 144*K2**2*K4**2 + 848*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1738*K2**2 + 296*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 876*K3**2 - 282*K4**2 - 32*K5**2 - 6*K6**2 + 1856
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.269']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11029', 'vk6.11107', 'vk6.12195', 'vk6.12302', 'vk6.14093', 'vk6.14098', 'vk6.14290', 'vk6.14301', 'vk6.15519', 'vk6.15532', 'vk6.16015', 'vk6.16022', 'vk6.16281', 'vk6.22574', 'vk6.22584', 'vk6.22737', 'vk6.22836', 'vk6.26112', 'vk6.26532', 'vk6.30606', 'vk6.30701', 'vk6.34049', 'vk6.34094', 'vk6.34524', 'vk6.34549', 'vk6.34571', 'vk6.38101', 'vk6.38138', 'vk6.42255', 'vk6.44621', 'vk6.44765', 'vk6.51838', 'vk6.54065', 'vk6.54510', 'vk6.54552', 'vk6.56574', 'vk6.56637', 'vk6.59011', 'vk6.64512', 'vk6.64521']
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 O1O2O3O4O5U1U3O6U5U6U4U2
R3 orbit {'O1O2O3O4O5U1U3O6U5U6U4U2', 'O1O2O3O4O5U1U3U4O6U5U6U2'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U4U2U6U1O6U3U5
Gauss code of K* O1O2O3O4U5U4U6U3U1O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U4U2U5U1U6
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 1 -1 2 1 1],[ 4 0 4 1 3 2 1],[-1 -4 0 -2 1 0 1],[ 1 -1 2 0 2 1 1],[-2 -3 -1 -2 0 -1 1],[-1 -2 0 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 1 -1 -4],[-2 0 1 -1 -1 -2 -3],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 0 -1 -2],[-1 1 1 0 0 -2 -4],[ 1 2 1 1 2 0 -1],[ 4 3 1 2 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,1,4,-1,1,1,2,3,1,1,1,1,0,1,2,2,4,1]
Phi over symmetry [-4,-1,1,1,1,2,1,1,2,4,3,1,1,2,2,-1,-1,-1,0,1,1]
Phi of -K [-4,-1,1,1,1,2,2,1,3,4,3,0,1,1,1,0,-1,0,-1,0,2]
Phi of K* [-2,-1,-1,-1,1,4,0,0,2,1,3,0,1,0,1,1,1,3,1,4,2]
Phi of -K* [-4,-1,1,1,1,2,1,1,2,4,3,1,1,2,2,-1,-1,-1,0,1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial 2w^3z^2+15w^2z+23w
Inner characteristic polynomial t^6+46t^4+17t^2
Outer characteristic polynomial t^7+70t^5+78t^3+3t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -128*K1**6 - 256*K1**4*K2**2 + 672*K1**4*K2 - 1104*K1**4 + 320*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**4 + 480*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 2272*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 3576*K1**2*K2 - 176*K1**2*K3**2 - 32*K1**2*K4**2 - 2236*K1**2 + 576*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 2872*K1*K2*K3 + 536*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 608*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 544*K2**2*K3**2 - 32*K2**2*K3*K7 - 144*K2**2*K4**2 + 848*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1738*K2**2 + 296*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 876*K3**2 - 282*K4**2 - 32*K5**2 - 6*K6**2 + 1856
Genus of based matrix 1
Fillings of based matrix [[{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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