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

Flat knot 6.808

Min(phi) over symmetries of the knot is: [-3,-2,-2,2,2,3,-1,0,2,3,4,0,0,1,2,1,3,3,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.808']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 4*K1*K2 - 4*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.570', '6.808', '6.1005', '6.1045', '6.1134', '6.1538', '6.1819']
Outer characteristic polynomial of the knot is: t^7+109t^5+165t^3+21t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.808']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 768*K1**4*K2 - 1696*K1**4 + 128*K1**3*K2*K3 + 64*K1**3*K3*K4 - 448*K1**3*K3 + 960*K1**2*K2**3 - 6816*K1**2*K2**2 - 384*K1**2*K2*K4 + 8976*K1**2*K2 - 256*K1**2*K3**2 - 64*K1**2*K3*K5 - 64*K1**2*K4**2 - 6248*K1**2 + 192*K1*K2**3*K3 - 384*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 6480*K1*K2*K3 + 704*K1*K3*K4 + 144*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1664*K2**4 - 192*K2**2*K3**2 - 48*K2**2*K4**2 + 1376*K2**2*K4 - 3704*K2**2 + 144*K2*K3*K5 + 32*K2*K4*K6 - 1736*K3**2 - 536*K4**2 - 80*K5**2 - 8*K6**2 + 4550
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.808']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71360', 'vk6.71420', 'vk6.71881', 'vk6.71942', 'vk6.74318', 'vk6.74963', 'vk6.76530', 'vk6.76937', 'vk6.77016', 'vk6.77073', 'vk6.77395', 'vk6.79372', 'vk6.79796', 'vk6.80829', 'vk6.81283', 'vk6.81484', 'vk6.83848', 'vk6.87076', 'vk6.88049', 'vk6.89567']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U1O5U2U6U4U5O6U3
R3 orbit {'O1O2O3O4U1O5U2U6U4U5O6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2O5U6U1U5U3O6U4
Gauss code of K* O1O2O3O4U2O5U6U1U5U3O6U4
Gauss code of -K* Same
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 -3 -2 2 2 3 -2],[ 3 0 1 3 2 2 2],[ 2 -1 0 3 1 2 0],[-2 -3 -3 0 0 2 -4],[-2 -2 -1 0 0 1 -3],[-3 -2 -2 -2 -1 0 -3],[ 2 -2 0 4 3 3 0]]
Primitive based matrix [[ 0 3 2 2 -2 -2 -3],[-3 0 -1 -2 -2 -3 -2],[-2 1 0 0 -1 -3 -2],[-2 2 0 0 -3 -4 -3],[ 2 2 1 3 0 0 -1],[ 2 3 3 4 0 0 -2],[ 3 2 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,2,2,3,1,2,2,3,2,0,1,3,2,3,4,3,0,1,2]
Phi over symmetry [-3,-2,-2,2,2,3,-1,0,2,3,4,0,0,1,2,1,3,3,0,-1,0]
Phi of -K [-3,-2,-2,2,2,3,-1,0,2,3,4,0,0,1,2,1,3,3,0,-1,0]
Phi of K* [-3,-2,-2,2,2,3,-1,0,2,3,4,0,0,1,2,1,3,3,0,-1,0]
Phi of -K* [-3,-2,-2,2,2,3,1,2,2,3,2,0,1,3,2,3,4,3,0,1,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2-4w^3z+23w^2z+31w
Inner characteristic polynomial t^6+75t^4+75t^2+9
Outer characteristic polynomial t^7+109t^5+165t^3+21t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 4*K1*K2 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**4*K2**2 + 768*K1**4*K2 - 1696*K1**4 + 128*K1**3*K2*K3 + 64*K1**3*K3*K4 - 448*K1**3*K3 + 960*K1**2*K2**3 - 6816*K1**2*K2**2 - 384*K1**2*K2*K4 + 8976*K1**2*K2 - 256*K1**2*K3**2 - 64*K1**2*K3*K5 - 64*K1**2*K4**2 - 6248*K1**2 + 192*K1*K2**3*K3 - 384*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 6480*K1*K2*K3 + 704*K1*K3*K4 + 144*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1664*K2**4 - 192*K2**2*K3**2 - 48*K2**2*K4**2 + 1376*K2**2*K4 - 3704*K2**2 + 144*K2*K3*K5 + 32*K2*K4*K6 - 1736*K3**2 - 536*K4**2 - 80*K5**2 - 8*K6**2 + 4550
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice True
Contact