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Flat knot 6.787

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,2,1,2,4,1,0,0,1,0,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.787']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 2*K1*K2 - 2*K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.618', '6.640', '6.736', '6.787']
Outer characteristic polynomial of the knot is: t^7+46t^5+49t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.787']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 1472*K1**4*K2 - 5008*K1**4 + 480*K1**3*K2*K3 - 1184*K1**3*K3 - 192*K1**2*K2**4 + 672*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5472*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 9824*K1**2*K2 - 1232*K1**2*K3**2 - 96*K1**2*K3*K5 - 80*K1**2*K4**2 - 4364*K1**2 + 128*K1*K2**3*K3 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6456*K1*K2*K3 + 1240*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 424*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 488*K2**2*K4 - 3728*K2**2 + 56*K2*K3*K5 - 1596*K3**2 - 298*K4**2 - 8*K5**2 + 3968
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.787']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4085', 'vk6.4118', 'vk6.5323', 'vk6.5356', 'vk6.7447', 'vk6.7478', 'vk6.8946', 'vk6.8979', 'vk6.10127', 'vk6.10292', 'vk6.10317', 'vk6.14538', 'vk6.15269', 'vk6.15396', 'vk6.15758', 'vk6.16173', 'vk6.29867', 'vk6.29900', 'vk6.33907', 'vk6.33990', 'vk6.34205', 'vk6.34371', 'vk6.48468', 'vk6.49170', 'vk6.50215', 'vk6.50244', 'vk6.51603', 'vk6.53966', 'vk6.54029', 'vk6.54178', 'vk6.54467', 'vk6.63314']
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 O1O2O3O4U3O5U6U5U1O6U2U4
R3 orbit {'O1O2O3O4U3O5U6U5U1O6U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U3O5U4U6U5O6U2
Gauss code of K* O1O2O3U1O4O5U3U4U6U5O6U2
Gauss code of -K* O1O2O3U2O4U5U4U6U1O5O6U3
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 -1 0 -1 3 1 -2],[ 1 0 0 0 2 0 0],[ 0 0 0 0 2 1 -1],[ 1 0 0 0 1 0 1],[-3 -2 -2 -1 0 1 -4],[-1 0 -1 0 -1 0 -1],[ 2 0 1 -1 4 1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -2 -1 -2 -4],[-1 -1 0 -1 0 0 -1],[ 0 2 1 0 0 0 -1],[ 1 1 0 0 0 0 1],[ 1 2 0 0 0 0 0],[ 2 4 1 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,-1,2,1,2,4,1,0,0,1,0,0,1,0,-1,0]
Phi over symmetry [-3,-1,0,1,1,2,-1,2,1,2,4,1,0,0,1,0,0,1,0,-1,0]
Phi of -K [-2,-1,-1,0,1,3,1,2,1,2,1,0,1,2,2,1,2,3,0,1,3]
Phi of K* [-3,-1,0,1,1,2,3,1,2,3,1,0,2,2,2,1,1,1,0,1,2]
Phi of -K* [-2,-1,-1,0,1,3,-1,0,1,1,4,0,0,0,1,0,0,2,1,2,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 2z^2+21z+35
Enhanced Jones-Krushkal polynomial 2w^3z^2+21w^2z+35w
Inner characteristic polynomial t^6+30t^4+28t^2+4
Outer characteristic polynomial t^7+46t^5+49t^3+9t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 2*K1*K2 - 2*K1 + 5*K2 + 6
2-strand cable arrow polynomial -64*K1**6 - 64*K1**4*K2**2 + 1472*K1**4*K2 - 5008*K1**4 + 480*K1**3*K2*K3 - 1184*K1**3*K3 - 192*K1**2*K2**4 + 672*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5472*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 9824*K1**2*K2 - 1232*K1**2*K3**2 - 96*K1**2*K3*K5 - 80*K1**2*K4**2 - 4364*K1**2 + 128*K1*K2**3*K3 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6456*K1*K2*K3 + 1240*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 424*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 488*K2**2*K4 - 3728*K2**2 + 56*K2*K3*K5 - 1596*K3**2 - 298*K4**2 - 8*K5**2 + 3968
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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