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

Flat knot 6.1994

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,0,0,0,0,1,1,1,0,0,-1,1,0,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1994', '7.44295']
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+10t^5+19t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1994', '7.44295']
2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 1152*K1**4*K2**2 + 3392*K1**4*K2 - 6880*K1**4 + 832*K1**3*K2*K3 - 256*K1**3*K3 + 1664*K1**2*K2**3 - 8512*K1**2*K2**2 - 256*K1**2*K2*K4 + 9312*K1**2*K2 - 128*K1**2*K3**2 - 192*K1**2 - 896*K1*K2**2*K3 + 4528*K1*K2*K3 + 48*K1*K3*K4 - 1136*K2**4 + 784*K2**2*K4 - 1776*K2**2 - 480*K3**2 - 52*K4**2 + 2178
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1994']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14', 'vk6.25', 'vk6.146', 'vk6.163', 'vk6.1201', 'vk6.1297', 'vk6.1309', 'vk6.2360', 'vk6.2398', 'vk6.2964', 'vk6.3524', 'vk6.6908', 'vk6.6941', 'vk6.15378', 'vk6.15499', 'vk6.33446', 'vk6.33503', 'vk6.33611', 'vk6.49937', 'vk6.53757']
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 O1O2U1O3U2O4U3U4O5O6U5U6
R3 orbit {'O1O2U1O3U2O4U3U4O5O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2U3U4O3O4U5U6O5U1O6U2
Gauss code of K* O1O2U3U4O3O4U5U6O5U1O6U2
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 -1 0 0 1 -1 1],[ 1 0 1 1 0 0 0],[ 0 -1 0 1 1 0 0],[ 0 -1 -1 0 1 0 0],[-1 0 -1 -1 0 0 0],[ 1 0 0 0 0 0 1],[-1 0 0 0 0 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 0 -1 -1 0 0],[ 0 0 1 0 1 -1 0],[ 0 0 1 -1 0 -1 0],[ 1 0 0 1 1 0 0],[ 1 1 0 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,0,0,0,0,1,1,1,0,0,-1,1,0,1,0,0]
Phi over symmetry [-1,-1,0,0,1,1,0,0,0,0,1,1,1,0,0,-1,1,0,1,0,0]
Phi of -K [-1,-1,0,0,1,1,0,0,0,2,2,1,1,1,2,-1,1,0,1,0,0]
Phi of K* [-1,-1,0,0,1,1,0,0,0,2,2,1,1,1,2,-1,1,0,1,0,0]
Phi of -K* [-1,-1,0,0,1,1,0,0,0,0,1,1,1,0,0,-1,1,0,1,0,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+6t^4+5t^2
Outer characteristic polynomial t^7+10t^5+19t^3+2t
Flat arrow polynomial -12*K1**2 + 6*K2 + 7
2-strand cable arrow polynomial -1024*K1**6 - 1152*K1**4*K2**2 + 3392*K1**4*K2 - 6880*K1**4 + 832*K1**3*K2*K3 - 256*K1**3*K3 + 1664*K1**2*K2**3 - 8512*K1**2*K2**2 - 256*K1**2*K2*K4 + 9312*K1**2*K2 - 128*K1**2*K3**2 - 192*K1**2 - 896*K1*K2**2*K3 + 4528*K1*K2*K3 + 48*K1*K3*K4 - 1136*K2**4 + 784*K2**2*K4 - 1776*K2**2 - 480*K3**2 - 52*K4**2 + 2178
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice True
Contact