Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 6.574

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,3,2,3,1,2,1,2,0,0,0,2,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.574']
Arrow polynomial of the knot is: -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.574', '6.593', '6.604', '6.649', '6.650', '6.673', '6.690', '6.783', '6.973', '6.985', '6.1033', '6.1035']
Outer characteristic polynomial of the knot is: t^7+67t^5+35t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.574']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 192*K1**4*K2 - 3216*K1**4 - 736*K1**3*K3 + 32*K1**2*K2**3 - 3616*K1**2*K2**2 - 288*K1**2*K2*K4 + 8232*K1**2*K2 - 464*K1**2*K3**2 - 48*K1**2*K4**2 - 4372*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2**2*K5 + 5304*K1*K2*K3 + 808*K1*K3*K4 + 64*K1*K4*K5 - 504*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 640*K2**2*K4 - 3446*K2**2 + 112*K2*K3*K5 + 8*K2*K4*K6 - 1544*K3**2 - 358*K4**2 - 44*K5**2 - 2*K6**2 + 3668
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.574']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16935', 'vk6.17176', 'vk6.20553', 'vk6.21954', 'vk6.23335', 'vk6.23628', 'vk6.28011', 'vk6.29478', 'vk6.35371', 'vk6.35790', 'vk6.39415', 'vk6.41608', 'vk6.42848', 'vk6.43125', 'vk6.45995', 'vk6.47671', 'vk6.55098', 'vk6.55353', 'vk6.57421', 'vk6.58592', 'vk6.59500', 'vk6.59794', 'vk6.62092', 'vk6.63070', 'vk6.64943', 'vk6.65149', 'vk6.66957', 'vk6.67818', 'vk6.68236', 'vk6.68377', 'vk6.69572', 'vk6.70269']
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 O1O2O3O4U1O5U2O6U4U6U5U3
R3 orbit {'O1O2O3O4U1O5U2O6U4U6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6U1O6U3O5U4
Gauss code of K* O1O2O3O4U5U6U4U1O5U3O6U2
Gauss code of -K* O1O2O3O4U3O5U2O6U4U1U5U6
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 -2 2 0 2 1],[ 3 0 1 3 2 2 1],[ 2 -1 0 3 1 2 1],[-2 -3 -3 0 -2 1 1],[ 0 -2 -1 2 0 2 1],[-2 -2 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 1 1 -2 -3 -3],[-2 -1 0 0 -2 -2 -2],[-1 -1 0 0 -1 -1 -1],[ 0 2 2 1 0 -1 -2],[ 2 3 2 1 1 0 -1],[ 3 3 2 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,-1,-1,2,3,3,0,2,2,2,1,1,1,1,2,1]
Phi over symmetry [-3,-2,0,1,2,2,0,1,3,2,3,1,2,1,2,0,0,0,2,1,-1]
Phi of -K [-3,-2,0,1,2,2,0,1,3,2,3,1,2,1,2,0,0,0,2,1,-1]
Phi of K* [-2,-2,-1,0,2,3,-1,1,0,2,3,2,0,1,2,0,2,3,1,1,0]
Phi of -K* [-3,-2,0,1,2,2,1,2,1,2,3,1,1,2,3,1,2,2,0,-1,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+45t^4+8t^2
Outer characteristic polynomial t^7+67t^5+35t^3+4t
Flat arrow polynomial -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -64*K1**6 + 192*K1**4*K2 - 3216*K1**4 - 736*K1**3*K3 + 32*K1**2*K2**3 - 3616*K1**2*K2**2 - 288*K1**2*K2*K4 + 8232*K1**2*K2 - 464*K1**2*K3**2 - 48*K1**2*K4**2 - 4372*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2**2*K5 + 5304*K1*K2*K3 + 808*K1*K3*K4 + 64*K1*K4*K5 - 504*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 640*K2**2*K4 - 3446*K2**2 + 112*K2*K3*K5 + 8*K2*K4*K6 - 1544*K3**2 - 358*K4**2 - 44*K5**2 - 2*K6**2 + 3668
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
Contact