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Flat knot 6.694

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,3,4,0,0,2,2,0,-1,0,-1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.694']
Arrow polynomial of the knot is: -12*K1**2 - 6*K1*K2 + 3*K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.694', '6.741']
Outer characteristic polynomial of the knot is: t^7+62t^5+171t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.694']
2-strand cable arrow polynomial of the knot is: -320*K1**6 - 320*K1**4*K2**2 + 832*K1**4*K2 - 2736*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 3248*K1**2*K2**2 + 4224*K1**2*K2 - 944*K1**2*K3**2 - 112*K1**2*K4**2 - 1252*K1**2 + 3656*K1*K2*K3 + 880*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 896*K2**4 - 496*K2**2*K3**2 - 24*K2**2*K4**2 + 736*K2**2*K4 - 1626*K2**2 + 480*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 1172*K3**2 - 428*K4**2 - 160*K5**2 - 38*K6**2 + 2226
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.694']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11272', 'vk6.11350', 'vk6.12537', 'vk6.12648', 'vk6.17606', 'vk6.18921', 'vk6.18999', 'vk6.19347', 'vk6.19640', 'vk6.24063', 'vk6.24155', 'vk6.25519', 'vk6.25620', 'vk6.26119', 'vk6.26537', 'vk6.30946', 'vk6.31069', 'vk6.32126', 'vk6.32245', 'vk6.36409', 'vk6.37654', 'vk6.37703', 'vk6.43511', 'vk6.44772', 'vk6.52022', 'vk6.52113', 'vk6.52937', 'vk6.56502', 'vk6.56661', 'vk6.65389', 'vk6.66124', 'vk6.66160']
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 O1O2O3O4U3O5U6U2O6U1U5U4
R3 orbit {'O1O2O3O4U3O5U6U2O6U1U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4O6U3U6O5U2
Gauss code of K* O1O2O3U1U4U5U3O5U2O6O4U6
Gauss code of -K* O1O2O3U4O5O4U2O6U1U6U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -1 -1 3 2 -1],[ 2 0 1 0 4 2 1],[ 1 -1 0 0 2 0 1],[ 1 0 0 0 1 0 1],[-3 -4 -2 -1 0 0 -3],[-2 -2 0 0 0 0 -2],[ 1 -1 -1 -1 3 2 0]]
Primitive based matrix [[ 0 3 2 -1 -1 -1 -2],[-3 0 0 -1 -2 -3 -4],[-2 0 0 0 0 -2 -2],[ 1 1 0 0 0 1 0],[ 1 2 0 0 0 1 -1],[ 1 3 2 -1 -1 0 -1],[ 2 4 2 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,1,1,1,2,0,1,2,3,4,0,0,2,2,0,-1,0,-1,1,1]
Phi over symmetry [-3,-2,1,1,1,2,0,1,2,3,4,0,0,2,2,0,-1,0,-1,1,1]
Phi of -K [-2,-1,-1,-1,2,3,0,0,1,2,1,-1,0,3,2,1,1,1,3,3,1]
Phi of K* [-3,-2,1,1,1,2,1,1,2,3,1,1,3,3,2,-1,-1,0,0,0,1]
Phi of -K* [-2,-1,-1,-1,2,3,0,1,1,2,4,0,1,0,1,1,0,2,2,3,0]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 14z+29
Enhanced Jones-Krushkal polynomial 14w^2z+29w
Inner characteristic polynomial t^6+42t^4+101t^2
Outer characteristic polynomial t^7+62t^5+171t^3
Flat arrow polynomial -12*K1**2 - 6*K1*K2 + 3*K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -320*K1**6 - 320*K1**4*K2**2 + 832*K1**4*K2 - 2736*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 3248*K1**2*K2**2 + 4224*K1**2*K2 - 944*K1**2*K3**2 - 112*K1**2*K4**2 - 1252*K1**2 + 3656*K1*K2*K3 + 880*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 896*K2**4 - 496*K2**2*K3**2 - 24*K2**2*K4**2 + 736*K2**2*K4 - 1626*K2**2 + 480*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 1172*K3**2 - 428*K4**2 - 160*K5**2 - 38*K6**2 + 2226
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}]]
If K is slice False
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