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

Flat knot 6.839

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,2,1,3,1,1,1,2,-1,1,0,2,2,-2]
Flat knots (up to 7 crossings) with same phi are :['6.839']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166']
Outer characteristic polynomial of the knot is: t^7+55t^5+58t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.839']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 64*K1**4*K2 - 96*K1**4 + 128*K1**3*K2*K3 - 96*K1**3*K3 - 384*K1**2*K2**4 + 992*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 3232*K1**2*K2**2 - 96*K1**2*K2*K4 + 2672*K1**2*K2 - 112*K1**2*K3**2 - 1944*K1**2 + 1184*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3**3 - 64*K1*K2*K3*K4 + 3080*K1*K2*K3 + 184*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1472*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 24*K2**2*K4**2 + 912*K2**2*K4 - 764*K2**2 - 32*K2*K3**2*K4 + 192*K2*K3*K5 + 24*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 808*K3**2 - 116*K4**2 - 24*K5**2 - 4*K6**2 + 1442
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.839']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70460', 'vk6.70475', 'vk6.70525', 'vk6.70600', 'vk6.70630', 'vk6.70655', 'vk6.70756', 'vk6.70842', 'vk6.70915', 'vk6.70943', 'vk6.71002', 'vk6.71105', 'vk6.71154', 'vk6.71169', 'vk6.71239', 'vk6.71298', 'vk6.71323', 'vk6.71338', 'vk6.73545', 'vk6.74217', 'vk6.74357', 'vk6.75006', 'vk6.75302', 'vk6.76414', 'vk6.76579', 'vk6.76641', 'vk6.76984', 'vk6.78286', 'vk6.79265', 'vk6.79397', 'vk6.79935', 'vk6.80754', 'vk6.81509', 'vk6.83986', 'vk6.86361', 'vk6.86871', 'vk6.87279', 'vk6.88073', 'vk6.88236', 'vk6.88244']
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 O1O2O3O4U1U2O5O6U4U5U6U3
R3 orbit {'O1O2O3O4U1U2O5U3O6U5U4U6', 'O1O2O3O4U1U2O5O6U4U5U6U3'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U2U5U6U1O5O6U3U4
Gauss code of K* O1O2O3O4U5U6U4U1O5O6U2U3
Gauss code of -K* O1O2O3O4U2U3O5O6U4U1U5U6
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 2 0 0 2],[ 3 0 1 3 2 1 1],[ 1 -1 0 2 1 1 1],[-2 -3 -2 0 -2 0 2],[ 0 -2 -1 2 0 1 2],[ 0 -1 -1 0 -1 0 1],[-2 -1 -1 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 2 0 -2 -2 -3],[-2 -2 0 -1 -2 -1 -1],[ 0 0 1 0 -1 -1 -1],[ 0 2 2 1 0 -1 -2],[ 1 2 1 1 1 0 -1],[ 3 3 1 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-2,0,2,2,3,1,2,1,1,1,1,1,1,2,1]
Phi over symmetry [-3,-1,0,0,2,2,1,1,2,1,3,1,1,1,2,-1,1,0,2,2,-2]
Phi of -K [-3,-1,0,0,2,2,1,1,2,2,4,0,0,1,2,-1,0,0,2,1,-2]
Phi of K* [-2,-2,0,0,1,3,-2,0,1,2,4,0,2,1,2,1,0,1,0,2,1]
Phi of -K* [-3,-1,0,0,2,2,1,1,2,1,3,1,1,1,2,-1,1,0,2,2,-2]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+18z+17
Enhanced Jones-Krushkal polynomial 5w^3z^2+18w^2z+17w
Inner characteristic polynomial t^6+37t^4+8t^2
Outer characteristic polynomial t^7+55t^5+58t^3+3t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -64*K1**4*K2**2 + 64*K1**4*K2 - 96*K1**4 + 128*K1**3*K2*K3 - 96*K1**3*K3 - 384*K1**2*K2**4 + 992*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 3232*K1**2*K2**2 - 96*K1**2*K2*K4 + 2672*K1**2*K2 - 112*K1**2*K3**2 - 1944*K1**2 + 1184*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3**3 - 64*K1*K2*K3*K4 + 3080*K1*K2*K3 + 184*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1472*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 24*K2**2*K4**2 + 912*K2**2*K4 - 764*K2**2 - 32*K2*K3**2*K4 + 192*K2*K3*K5 + 24*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 808*K3**2 - 116*K4**2 - 24*K5**2 - 4*K6**2 + 1442
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
Contact