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

Flat knot 6.1468

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,1,4,0,1,0,1,1,1,1,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1468']
Arrow polynomial of the knot is: -12*K1**2 + 6*K2 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.612', '6.1246', '6.1402', '6.1410', '6.1411', '6.1468', '6.1617', '6.1666', '6.1667', '6.1815', '6.1904', '6.1994', '6.1995', '6.1996', '6.2001', '6.2014', '6.2015', '6.2016', '6.2017', '6.2020', '6.2022']
Outer characteristic polynomial of the knot is: t^7+37t^5+25t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1468']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1088*K1**4*K2 - 3424*K1**4 + 192*K1**3*K2*K3 + 128*K1**2*K2**3 - 2464*K1**2*K2**2 + 5280*K1**2*K2 - 160*K1**2*K3**2 - 2056*K1**2 + 2128*K1*K2*K3 + 176*K1*K3*K4 - 112*K2**4 + 160*K2**2*K4 - 2200*K2**2 - 632*K3**2 - 124*K4**2 + 2274
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1468']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3574', 'vk6.3606', 'vk6.3827', 'vk6.3860', 'vk6.6993', 'vk6.7026', 'vk6.7211', 'vk6.7237', 'vk6.15333', 'vk6.15458', 'vk6.33970', 'vk6.34014', 'vk6.34426', 'vk6.48230', 'vk6.48383', 'vk6.49960', 'vk6.49984', 'vk6.53998', 'vk6.54052', 'vk6.54498']
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 O1O2O3U1O4U3O5U4U6U5O6U2
R3 orbit {'O1O2O3U1O4U3O5U4U6U5O6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2O4U5U4U6O5U1O6U3
Gauss code of K* O1O2O3U2O4U5U4U6O5U1O6U3
Gauss code of -K* Same
Diagrammatic symmetry type -
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 0 2 -1],[ 2 0 2 1 1 0 2],[-1 -2 0 -1 0 2 -2],[ 0 -1 1 0 1 1 0],[ 0 -1 0 -1 0 1 -1],[-2 0 -2 -1 -1 0 -2],[ 1 -2 2 0 1 2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -1 -2 0],[-1 2 0 0 -1 -2 -2],[ 0 1 0 0 -1 -1 -1],[ 0 1 1 1 0 0 -1],[ 1 2 2 1 0 0 -2],[ 2 0 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,1,1,2,0,0,1,2,2,1,1,1,0,1,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,1,4,0,1,0,1,1,1,1,0,1,-1]
Phi of -K [-2,-1,0,0,1,2,-1,1,1,1,4,0,1,0,1,1,1,1,0,1,-1]
Phi of K* [-2,-1,0,0,1,2,-1,1,1,1,4,0,1,0,1,1,1,1,0,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,1,2,0,0,1,2,2,1,1,1,0,1,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 16z+33
Enhanced Jones-Krushkal polynomial 16w^2z+33w
Inner characteristic polynomial t^6+27t^4+5t^2
Outer characteristic polynomial t^7+37t^5+25t^3
Flat arrow polynomial -12*K1**2 + 6*K2 + 7
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1088*K1**4*K2 - 3424*K1**4 + 192*K1**3*K2*K3 + 128*K1**2*K2**3 - 2464*K1**2*K2**2 + 5280*K1**2*K2 - 160*K1**2*K3**2 - 2056*K1**2 + 2128*K1*K2*K3 + 176*K1*K3*K4 - 112*K2**4 + 160*K2**2*K4 - 2200*K2**2 - 632*K3**2 - 124*K4**2 + 2274
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice True
Contact