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

Flat knot 6.1746

Min(phi) over symmetries of the knot is: [-2,0,1,1,1,0,2,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1746', '6.1760', '7.43063']
Arrow polynomial of the knot is: -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.425', '6.655', '6.755', '6.769', '6.792', '6.1240', '6.1494', '6.1522', '6.1534', '6.1587', '6.1707', '6.1746', '6.1747', '6.1786', '6.1814', '6.1828', '6.1835', '6.1854', '6.1870']
Outer characteristic polynomial of the knot is: t^5+19t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1746', '6.1760', '7.37934']
2-strand cable arrow polynomial of the knot is: -704*K1**6 - 384*K1**4*K2**2 + 2048*K1**4*K2 - 4992*K1**4 + 832*K1**3*K2*K3 - 960*K1**3*K3 - 4208*K1**2*K2**2 - 672*K1**2*K2*K4 + 8944*K1**2*K2 - 1216*K1**2*K3**2 - 512*K1**2*K4**2 - 4812*K1**2 - 576*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 7304*K1*K2*K3 + 2680*K1*K3*K4 + 576*K1*K4*K5 - 296*K2**4 - 336*K2**2*K3**2 - 112*K2**2*K4**2 + 1224*K2**2*K4 - 4892*K2**2 - 96*K2*K3**2*K4 + 416*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2856*K3**2 - 1290*K4**2 - 212*K5**2 - 28*K6**2 + 5256
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1746']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4138', 'vk6.4169', 'vk6.5376', 'vk6.5407', 'vk6.7506', 'vk6.7531', 'vk6.9007', 'vk6.9038', 'vk6.12434', 'vk6.12467', 'vk6.13354', 'vk6.13579', 'vk6.13610', 'vk6.14246', 'vk6.14695', 'vk6.14735', 'vk6.15190', 'vk6.15849', 'vk6.15889', 'vk6.30847', 'vk6.30880', 'vk6.32031', 'vk6.32064', 'vk6.33072', 'vk6.33103', 'vk6.33849', 'vk6.34309', 'vk6.48494', 'vk6.50279', 'vk6.53542', 'vk6.53928', 'vk6.54254']
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 O1O2O3U4U2O4O5U1U5O6U3U6
R3 orbit {'O1O2O3U4U2O4O5U1U5O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O4U5U3O5O6U2U6
Gauss code of K* O1O2U3O4O3U1U5U4O6O5U6U2
Gauss code of -K* O1O2U1O3O4U3U5O6O5U2U6U4
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 -2 0 1 -1 1 1],[ 2 0 1 3 1 1 1],[ 0 -1 0 0 0 0 1],[-1 -3 0 0 -1 0 1],[ 1 -1 0 1 0 1 1],[-1 -1 0 0 -1 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 -2],[-1 0 1 0 -3],[-1 -1 0 -1 -1],[ 0 0 1 0 -1],[ 2 3 1 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,0,2,-1,0,3,1,1,1]
Phi over symmetry [-2,0,1,1,1,0,2,1,0,-1]
Phi of -K [-2,0,1,1,1,0,2,1,0,-1]
Phi of K* [-1,-1,0,2,-1,0,2,1,0,1]
Phi of -K* [-2,0,1,1,1,1,3,1,0,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^4+13t^2+4
Outer characteristic polynomial t^5+19t^3+7t
Flat arrow polynomial -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -704*K1**6 - 384*K1**4*K2**2 + 2048*K1**4*K2 - 4992*K1**4 + 832*K1**3*K2*K3 - 960*K1**3*K3 - 4208*K1**2*K2**2 - 672*K1**2*K2*K4 + 8944*K1**2*K2 - 1216*K1**2*K3**2 - 512*K1**2*K4**2 - 4812*K1**2 - 576*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 7304*K1*K2*K3 + 2680*K1*K3*K4 + 576*K1*K4*K5 - 296*K2**4 - 336*K2**2*K3**2 - 112*K2**2*K4**2 + 1224*K2**2*K4 - 4892*K2**2 - 96*K2*K3**2*K4 + 416*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2856*K3**2 - 1290*K4**2 - 212*K5**2 - 28*K6**2 + 5256
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
Contact