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Flat knot 6.983

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,3,4,2,0,2,1,1,1,0,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.983']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.576', '6.581', '6.622', '6.627', '6.983', '6.1017']
Outer characteristic polynomial of the knot is: t^7+40t^5+59t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.983']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 3264*K1**4*K2 - 4208*K1**4 + 128*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1248*K1**3*K3 - 448*K1**2*K2**4 + 2496*K1**2*K2**3 - 8400*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 11856*K1**2*K2 - 496*K1**2*K3**2 - 64*K1**2*K3*K5 - 96*K1**2*K4**2 - 7280*K1**2 + 384*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8744*K1*K2*K3 - 32*K1*K2*K4*K5 + 1264*K1*K3*K4 + 128*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1544*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 40*K2**2*K4**2 + 1912*K2**2*K4 - 5458*K2**2 + 352*K2*K3*K5 + 32*K2*K4*K6 - 2416*K3**2 - 642*K4**2 - 80*K5**2 - 6*K6**2 + 5744
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.983']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4408', 'vk6.4503', 'vk6.5794', 'vk6.5921', 'vk6.7861', 'vk6.7966', 'vk6.9281', 'vk6.9400', 'vk6.10156', 'vk6.10229', 'vk6.10374', 'vk6.17889', 'vk6.17952', 'vk6.18291', 'vk6.18628', 'vk6.24396', 'vk6.25182', 'vk6.30043', 'vk6.30106', 'vk6.36909', 'vk6.37369', 'vk6.43823', 'vk6.44130', 'vk6.44454', 'vk6.48615', 'vk6.50520', 'vk6.50603', 'vk6.51127', 'vk6.51667', 'vk6.55850', 'vk6.56078', 'vk6.65511']
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 O1O2O3O4U1U4O5U2O6U5U6U3
R3 orbit {'O1O2O3O4U1U4O5U2O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6O5U3O6U1U4
Gauss code of K* O1O2O3U4U5U3U6O4O6U1O5U2
Gauss code of -K* O1O2O3U2O4U3O5O6U5U1U4U6
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 -1 2 1 0 1],[ 3 0 2 3 1 1 0],[ 1 -2 0 2 0 1 1],[-2 -3 -2 0 0 -1 1],[-1 -1 0 0 0 0 0],[ 0 -1 -1 1 0 0 1],[-1 0 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 0 -1 -2 -3],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 1 2 1 0 1 0 -2],[ 3 3 0 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,0,1,2,3,0,1,1,0,0,0,1,1,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,3,4,2,0,2,1,1,1,0,1,0,1,2]
Phi of -K [-3,-1,0,1,1,2,0,2,3,4,2,0,2,1,1,1,0,1,0,1,2]
Phi of K* [-2,-1,-1,0,1,3,1,2,1,1,2,0,1,2,3,0,1,4,0,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,0,1,3,1,1,0,2,1,0,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+24t^4+24t^2
Outer characteristic polynomial t^7+40t^5+59t^3+7t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 3264*K1**4*K2 - 4208*K1**4 + 128*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1248*K1**3*K3 - 448*K1**2*K2**4 + 2496*K1**2*K2**3 - 8400*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 11856*K1**2*K2 - 496*K1**2*K3**2 - 64*K1**2*K3*K5 - 96*K1**2*K4**2 - 7280*K1**2 + 384*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8744*K1*K2*K3 - 32*K1*K2*K4*K5 + 1264*K1*K3*K4 + 128*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1544*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 40*K2**2*K4**2 + 1912*K2**2*K4 - 5458*K2**2 + 352*K2*K3*K5 + 32*K2*K4*K6 - 2416*K3**2 - 642*K4**2 - 80*K5**2 - 6*K6**2 + 5744
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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