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Flat knot 6.265

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,2,4,2,3,1,2,2,2,1,2,1,2,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.265']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 6*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.265', '6.495']
Outer characteristic polynomial of the knot is: t^7+85t^5+95t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.265']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 896*K1**4*K2 - 1440*K1**4 + 608*K1**3*K2*K3 - 288*K1**3*K3 - 640*K1**2*K2**4 + 2016*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 8624*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 800*K1**2*K2*K4 + 7592*K1**2*K2 - 480*K1**2*K3**2 - 64*K1**2*K3*K5 - 4404*K1**2 + 3392*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 2976*K1*K2**2*K3 - 736*K1*K2**2*K5 + 64*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8912*K1*K2*K3 - 64*K1*K2*K4*K5 + 1176*K1*K3*K4 + 104*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 448*K2**4*K4 - 3512*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 2784*K2**2*K3**2 - 336*K2**2*K4**2 + 2984*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2532*K2**2 - 64*K2*K3**2*K4 + 1352*K2*K3*K5 + 128*K2*K4*K6 + 8*K2*K5*K7 - 2148*K3**2 - 584*K4**2 - 136*K5**2 - 12*K6**2 + 3710
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.265']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16342', 'vk6.16383', 'vk6.18067', 'vk6.18405', 'vk6.22681', 'vk6.22758', 'vk6.24510', 'vk6.24933', 'vk6.34621', 'vk6.34702', 'vk6.36647', 'vk6.37071', 'vk6.42312', 'vk6.42341', 'vk6.43929', 'vk6.44248', 'vk6.54605', 'vk6.54642', 'vk6.55887', 'vk6.56175', 'vk6.59091', 'vk6.59128', 'vk6.60407', 'vk6.60766', 'vk6.64634', 'vk6.64678', 'vk6.65517', 'vk6.65833', 'vk6.67991', 'vk6.68015', 'vk6.68603', 'vk6.68820']
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 O1O2O3O4O5U1U3O6U4U5U6U2
R3 orbit {'O1O2O3O4O5U1U3O6U4U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U1U2O6U3U5
Gauss code of K* O1O2O3O4U5U4U6U1U2O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U3U4U5U1U6
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 0 2 2],[ 4 0 4 1 2 3 2],[-1 -4 0 -2 -1 1 2],[ 1 -1 2 0 1 2 2],[ 0 -2 1 -1 0 1 2],[-2 -3 -1 -2 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -1 -4],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -2 -2 -2 -2],[-1 1 2 0 -1 -2 -4],[ 0 1 2 1 0 -1 -2],[ 1 2 2 2 1 0 -1],[ 4 3 2 4 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,1,4,-1,1,1,2,3,2,2,2,2,1,2,4,1,2,1]
Phi over symmetry [-4,-1,0,1,2,2,1,2,4,2,3,1,2,2,2,1,2,1,2,1,-1]
Phi of -K [-4,-1,0,1,2,2,2,2,1,3,4,0,0,1,1,0,1,0,0,-1,-1]
Phi of K* [-2,-2,-1,0,1,4,-1,-1,0,1,4,0,1,1,3,0,0,1,0,2,2]
Phi of -K* [-4,-1,0,1,2,2,1,2,4,2,3,1,2,2,2,1,2,1,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+59t^4+19t^2+1
Outer characteristic polynomial t^7+85t^5+95t^3+7t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 6*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -384*K1**4*K2**2 + 896*K1**4*K2 - 1440*K1**4 + 608*K1**3*K2*K3 - 288*K1**3*K3 - 640*K1**2*K2**4 + 2016*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 8624*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 800*K1**2*K2*K4 + 7592*K1**2*K2 - 480*K1**2*K3**2 - 64*K1**2*K3*K5 - 4404*K1**2 + 3392*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 2976*K1*K2**2*K3 - 736*K1*K2**2*K5 + 64*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8912*K1*K2*K3 - 64*K1*K2*K4*K5 + 1176*K1*K3*K4 + 104*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 448*K2**4*K4 - 3512*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 2784*K2**2*K3**2 - 336*K2**2*K4**2 + 2984*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2532*K2**2 - 64*K2*K3**2*K4 + 1352*K2*K3*K5 + 128*K2*K4*K6 + 8*K2*K5*K7 - 2148*K3**2 - 584*K4**2 - 136*K5**2 - 12*K6**2 + 3710
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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