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Flat knot 6.279

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,1,4,4,3,1,3,2,2,1,1,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.279']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.153', '6.279']
Outer characteristic polynomial of the knot is: t^7+87t^5+93t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.279']
2-strand cable arrow polynomial of the knot is: 448*K1**4*K2 - 912*K1**4 - 640*K1**3*K2**2*K3 + 1408*K1**3*K2*K3 - 736*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 2720*K1**2*K2**3 - 768*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 8736*K1**2*K2**2 + 512*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 960*K1**2*K2*K4 + 7696*K1**2*K2 - 944*K1**2*K3**2 - 5156*K1**2 - 256*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2912*K1*K2**3*K3 + 800*K1*K2**2*K3*K4 - 2784*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 + 64*K1*K2*K3**3 - 800*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8528*K1*K2*K3 + 1208*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 480*K2**4*K4 - 3128*K2**4 + 288*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 2128*K2**2*K3**2 - 520*K2**2*K4**2 + 2424*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K6**2 - 2542*K2**2 - 32*K2*K3**2*K4 + 1080*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 2152*K3**2 - 536*K4**2 - 156*K5**2 - 10*K6**2 + 3862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.279']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16333', 'vk6.16376', 'vk6.18071', 'vk6.18408', 'vk6.22664', 'vk6.22747', 'vk6.24518', 'vk6.24939', 'vk6.34608', 'vk6.34687', 'vk6.36651', 'vk6.37078', 'vk6.42303', 'vk6.42334', 'vk6.43933', 'vk6.44252', 'vk6.54596', 'vk6.54637', 'vk6.55899', 'vk6.56184', 'vk6.59075', 'vk6.59120', 'vk6.60427', 'vk6.60781', 'vk6.64627', 'vk6.64667', 'vk6.65533', 'vk6.65846', 'vk6.67986', 'vk6.68010', 'vk6.68615', 'vk6.68830']
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 O1O2O3O4O5U1U5O6U2U3U6U4
R3 orbit {'O1O2O3O4O5U1U5O6U2U3U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U3U4O6U1U5
Gauss code of K* O1O2O3O4U5U1U2U4U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U1U3U4U6
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 -2 0 3 1 2],[ 4 0 2 3 4 1 2],[ 2 -2 0 1 3 0 2],[ 0 -3 -1 0 2 0 1],[-3 -4 -3 -2 0 0 0],[-1 -1 0 0 0 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 0 0 -1],[ 0 2 1 0 0 -1 -3],[ 2 3 2 0 1 0 -2],[ 4 4 2 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,0,0,2,3,4,0,1,2,2,0,0,1,1,3,2]
Phi over symmetry [-4,-2,0,1,2,3,0,1,4,4,3,1,3,2,2,1,1,1,1,2,1]
Phi of -K [-4,-2,0,1,2,3,0,1,4,4,3,1,3,2,2,1,1,1,1,2,1]
Phi of K* [-3,-2,-1,0,2,4,1,2,1,2,3,1,1,2,4,1,3,4,1,1,0]
Phi of -K* [-4,-2,0,1,2,3,2,3,1,2,4,1,0,2,3,0,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+53t^4+29t^2+1
Outer characteristic polynomial t^7+87t^5+93t^3+11t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial 448*K1**4*K2 - 912*K1**4 - 640*K1**3*K2**2*K3 + 1408*K1**3*K2*K3 - 736*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 2720*K1**2*K2**3 - 768*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 8736*K1**2*K2**2 + 512*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 960*K1**2*K2*K4 + 7696*K1**2*K2 - 944*K1**2*K3**2 - 5156*K1**2 - 256*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2912*K1*K2**3*K3 + 800*K1*K2**2*K3*K4 - 2784*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 + 64*K1*K2*K3**3 - 800*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8528*K1*K2*K3 + 1208*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 480*K2**4*K4 - 3128*K2**4 + 288*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 2128*K2**2*K3**2 - 520*K2**2*K4**2 + 2424*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K6**2 - 2542*K2**2 - 32*K2*K3**2*K4 + 1080*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 2152*K3**2 - 536*K4**2 - 156*K5**2 - 10*K6**2 + 3862
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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