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

Flat knot 6.1045

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,-1,2,1,3,1,0,0,0,1,1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1045']
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+47t^5+236t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1045']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 256*K1**4*K2**3 - 832*K1**4*K2**2 + 1088*K1**4*K2 - 2688*K1**4 + 352*K1**3*K2*K3 - 960*K1**2*K2**4 + 1632*K1**2*K2**3 - 5088*K1**2*K2**2 + 5592*K1**2*K2 - 128*K1**2*K3**2 - 1956*K1**2 + 832*K1*K2**3*K3 + 2840*K1*K2*K3 + 128*K1*K3*K4 - 192*K2**6 + 128*K2**4*K4 - 1280*K2**4 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 384*K2**2*K4 - 952*K2**2 + 48*K2*K3*K5 - 556*K3**2 - 100*K4**2 + 2026
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1045']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4687', 'vk6.4992', 'vk6.6169', 'vk6.6642', 'vk6.8166', 'vk6.8586', 'vk6.9556', 'vk6.9897', 'vk6.17387', 'vk6.20921', 'vk6.20981', 'vk6.22333', 'vk6.22405', 'vk6.23556', 'vk6.23895', 'vk6.28401', 'vk6.36147', 'vk6.40059', 'vk6.40182', 'vk6.42112', 'vk6.43060', 'vk6.43366', 'vk6.46591', 'vk6.46691', 'vk6.48727', 'vk6.49519', 'vk6.49724', 'vk6.51429', 'vk6.55537', 'vk6.58921', 'vk6.65283', 'vk6.69769']
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 O1O2O3O4U5U6O5U1O6U3U4U2
R3 orbit {'O1O2O3O4U5U6O5U1O6U3U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U2O5U4O6U5U6
Gauss code of K* O1O2O3U4U3U1U2O5O6U5O4U6
Gauss code of -K* O1O2O3U4O5U6O4O6U2U3U1U5
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 0 2 -1 0],[ 2 0 2 0 1 2 3],[-1 -2 0 -1 1 -2 0],[ 0 0 1 0 1 -1 1],[-2 -1 -1 -1 0 -3 -1],[ 1 -2 2 1 3 0 0],[ 0 -3 0 -1 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -1 -3 -1],[-1 1 0 0 -1 -2 -2],[ 0 1 0 0 -1 0 -3],[ 0 1 1 1 0 -1 0],[ 1 3 2 0 1 0 -2],[ 2 1 2 3 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,1,3,1,0,1,2,2,1,0,3,1,0,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,-1,2,1,3,1,0,0,0,1,1,1,0,1,0]
Phi of -K [-2,-1,0,0,1,2,-1,-1,2,1,3,1,0,0,0,1,1,1,0,1,0]
Phi of K* [-2,-1,0,0,1,2,0,1,1,0,3,0,1,0,1,1,0,2,1,-1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,0,3,2,1,1,0,2,3,1,1,1,0,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial -6w^3z+19w^2z+27w
Inner characteristic polynomial t^6+37t^4+174t^2
Outer characteristic polynomial t^7+47t^5+236t^3
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 4*K1*K2 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial -256*K1**6 + 256*K1**4*K2**3 - 832*K1**4*K2**2 + 1088*K1**4*K2 - 2688*K1**4 + 352*K1**3*K2*K3 - 960*K1**2*K2**4 + 1632*K1**2*K2**3 - 5088*K1**2*K2**2 + 5592*K1**2*K2 - 128*K1**2*K3**2 - 1956*K1**2 + 832*K1*K2**3*K3 + 2840*K1*K2*K3 + 128*K1*K3*K4 - 192*K2**6 + 128*K2**4*K4 - 1280*K2**4 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 384*K2**2*K4 - 952*K2**2 + 48*K2*K3*K5 - 556*K3**2 - 100*K4**2 + 2026
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
Contact