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Flat knot 6.464

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,0,1,3,1,3,0,1,0,1,1,0,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.464']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + 4*K2 + 2*K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.464']
Outer characteristic polynomial of the knot is: t^7+68t^5+128t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.464']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 864*K1**4*K2 - 3776*K1**4 + 608*K1**3*K2*K3 + 192*K1**3*K3*K4 - 864*K1**3*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5712*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 10352*K1**2*K2 - 1920*K1**2*K3**2 - 64*K1**2*K3*K5 - 352*K1**2*K4**2 - 6712*K1**2 + 704*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 + 192*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9944*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 64*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2800*K1*K3*K4 + 440*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1272*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1200*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 336*K2**2*K4**2 - 32*K2**2*K4*K8 + 2056*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 5548*K2**2 - 160*K2*K3**2*K4 + 992*K2*K3*K5 + 272*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 224*K3**4 - 80*K3**2*K4**2 + 176*K3**2*K6 - 3100*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1128*K4**2 - 212*K5**2 - 52*K6**2 - 8*K7**2 - 2*K8**2 + 5944
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.464']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4827', 'vk6.5172', 'vk6.6393', 'vk6.6826', 'vk6.8358', 'vk6.8790', 'vk6.9728', 'vk6.10033', 'vk6.11623', 'vk6.11974', 'vk6.12969', 'vk6.20452', 'vk6.20745', 'vk6.21806', 'vk6.27839', 'vk6.29348', 'vk6.31426', 'vk6.32604', 'vk6.39268', 'vk6.39777', 'vk6.41448', 'vk6.46337', 'vk6.47577', 'vk6.47914', 'vk6.49066', 'vk6.49900', 'vk6.51326', 'vk6.51545', 'vk6.53218', 'vk6.57323', 'vk6.62012', 'vk6.64299']
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 O1O2O3O4O5U6U4O6U1U3U5U2
R3 orbit {'O1O2O3O4O5U6U4O6U1U3U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U1U3U5O6U2U6
Gauss code of K* O1O2O3O4U1U4U2U5U3O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U2U6U3U1U4
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 0 0 3 -1],[ 3 0 3 1 1 3 2],[-1 -3 0 -1 0 2 -2],[ 0 -1 1 0 1 2 -1],[ 0 -1 0 -1 0 0 0],[-3 -3 -2 -2 0 0 -3],[ 1 -2 2 1 0 3 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 -2 0 -2 -3 -3],[-1 2 0 0 -1 -2 -3],[ 0 0 0 0 -1 0 -1],[ 0 2 1 1 0 -1 -1],[ 1 3 2 0 1 0 -2],[ 3 3 3 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,2,0,2,3,3,0,1,2,3,1,0,1,1,1,2]
Phi over symmetry [-3,-1,0,0,1,3,0,1,3,1,3,0,1,0,1,1,0,2,1,2,0]
Phi of -K [-3,-1,0,0,1,3,0,2,2,1,3,0,1,0,1,-1,0,1,1,3,0]
Phi of K* [-3,-1,0,0,1,3,0,1,3,1,3,0,1,0,1,1,0,2,1,2,0]
Phi of -K* [-3,-1,0,0,1,3,2,1,1,3,3,0,1,2,3,-1,0,0,1,2,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+23z+35
Enhanced Jones-Krushkal polynomial 3w^3z^2+23w^2z+35w
Inner characteristic polynomial t^6+48t^4+76t^2+4
Outer characteristic polynomial t^7+68t^5+128t^3+10t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + 4*K2 + 2*K3 + K4 + 6
2-strand cable arrow polynomial -128*K1**6 + 864*K1**4*K2 - 3776*K1**4 + 608*K1**3*K2*K3 + 192*K1**3*K3*K4 - 864*K1**3*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5712*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 10352*K1**2*K2 - 1920*K1**2*K3**2 - 64*K1**2*K3*K5 - 352*K1**2*K4**2 - 6712*K1**2 + 704*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 + 192*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9944*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 64*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2800*K1*K3*K4 + 440*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1272*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1200*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 336*K2**2*K4**2 - 32*K2**2*K4*K8 + 2056*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 5548*K2**2 - 160*K2*K3**2*K4 + 992*K2*K3*K5 + 272*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 224*K3**4 - 80*K3**2*K4**2 + 176*K3**2*K6 - 3100*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1128*K4**2 - 212*K5**2 - 52*K6**2 - 8*K7**2 - 2*K8**2 + 5944
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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