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Flat knot 6.832

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,0,1,3,2,3,1,3,2,3,0,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.832']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+76t^5+28t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.832']
2-strand cable arrow polynomial of the knot is: 3264*K1**4*K2 - 5312*K1**4 + 1536*K1**3*K2*K3 - 1536*K1**3*K3 - 128*K1**2*K2**4 + 1344*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 8576*K1**2*K2**2 - 1408*K1**2*K2*K4 + 8576*K1**2*K2 - 1344*K1**2*K3**2 - 96*K1**2*K4**2 - 2672*K1**2 + 768*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 576*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 7328*K1*K2*K3 + 1968*K1*K3*K4 + 288*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 1360*K2**4 - 64*K2**3*K6 - 512*K2**2*K3**2 - 112*K2**2*K4**2 + 1640*K2**2*K4 - 2758*K2**2 + 608*K2*K3*K5 + 72*K2*K4*K6 - 1680*K3**2 - 752*K4**2 - 176*K5**2 - 10*K6**2 + 3214
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.832']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20143', 'vk6.20169', 'vk6.21433', 'vk6.21453', 'vk6.27255', 'vk6.27297', 'vk6.28915', 'vk6.28954', 'vk6.38676', 'vk6.38726', 'vk6.40904', 'vk6.47262', 'vk6.47295', 'vk6.56976', 'vk6.57003', 'vk6.58128', 'vk6.62677', 'vk6.67469', 'vk6.70035', 'vk6.70052']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4U3O5U2U1U6U4O6U5
R3 orbit {'O1O2O3O4U3O5U2U1U6U4O6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5O6U1U6U4U3O5U2
Gauss code of K* Same
Gauss code of -K* O1O2O3O4U5O6U1U6U4U3O5U2
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -2 -1 3 3 -1],[ 2 0 0 0 3 3 1],[ 2 0 0 0 2 2 1],[ 1 0 0 0 1 1 0],[-3 -3 -2 -1 0 0 -3],[-3 -3 -2 -1 0 0 -3],[ 1 -1 -1 0 3 3 0]]
Primitive based matrix [[ 0 3 3 -1 -1 -2 -2],[-3 0 0 -1 -3 -2 -3],[-3 0 0 -1 -3 -2 -3],[ 1 1 1 0 0 0 0],[ 1 3 3 0 0 -1 -1],[ 2 2 2 0 1 0 0],[ 2 3 3 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,1,1,2,2,0,1,3,2,3,1,3,2,3,0,0,0,1,1,0]
Phi over symmetry [-3,-3,1,1,2,2,0,1,3,2,3,1,3,2,3,0,0,0,1,1,0]
Phi of -K [-2,-2,-1,-1,3,3,0,0,1,2,2,0,1,3,3,0,1,1,3,3,0]
Phi of K* [-3,-3,1,1,2,2,0,1,3,2,3,1,3,2,3,0,0,0,1,1,0]
Phi of -K* [-2,-2,-1,-1,3,3,0,0,1,2,2,0,1,3,3,0,1,1,3,3,0]
Symmetry type of based matrix +
u-polynomial -2t^3+2t^2+2t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+48t^4+6t^2
Outer characteristic polynomial t^7+76t^5+28t^3+2t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial 3264*K1**4*K2 - 5312*K1**4 + 1536*K1**3*K2*K3 - 1536*K1**3*K3 - 128*K1**2*K2**4 + 1344*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 8576*K1**2*K2**2 - 1408*K1**2*K2*K4 + 8576*K1**2*K2 - 1344*K1**2*K3**2 - 96*K1**2*K4**2 - 2672*K1**2 + 768*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 576*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 7328*K1*K2*K3 + 1968*K1*K3*K4 + 288*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 1360*K2**4 - 64*K2**3*K6 - 512*K2**2*K3**2 - 112*K2**2*K4**2 + 1640*K2**2*K4 - 2758*K2**2 + 608*K2*K3*K5 + 72*K2*K4*K6 - 1680*K3**2 - 752*K4**2 - 176*K5**2 - 10*K6**2 + 3214
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice False
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