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

Flat knot 6.1967

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,1,1,3,0,-1,0,0,0,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1967']
Arrow polynomial of the knot is: -6*K1**2 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.689', '6.691', '6.752', '6.754', '6.1106', '6.1116', '6.1126', '6.1335', '6.1379', '6.1386', '6.1409', '6.1415', '6.1417', '6.1418', '6.1421', '6.1422', '6.1428', '6.1431', '6.1432', '6.1435', '6.1443', '6.1445', '6.1446', '6.1447', '6.1454', '6.1455', '6.1460', '6.1462', '6.1464', '6.1466', '6.1472', '6.1474', '6.1475', '6.1501', '6.1516', '6.1518', '6.1566', '6.1570', '6.1590', '6.1599', '6.1602', '6.1603', '6.1604', '6.1605', '6.1614', '6.1615', '6.1625', '6.1628', '6.1730', '6.1780', '6.1883', '6.1885', '6.1888', '6.1890', '6.1941', '6.1943', '6.1945', '6.1948', '6.1961', '6.1963', '6.1966', '6.1967', '6.1971']
Outer characteristic polynomial of the knot is: t^7+25t^5+55t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1967']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 1248*K1**4*K2 - 3696*K1**4 + 32*K1**3*K2*K3 - 576*K1**3*K3 + 352*K1**2*K2**3 - 4448*K1**2*K2**2 - 608*K1**2*K2*K4 + 7896*K1**2*K2 - 112*K1**2*K3**2 - 4524*K1**2 - 576*K1*K2**2*K3 + 6056*K1*K2*K3 + 1080*K1*K3*K4 - 152*K2**4 + 688*K2**2*K4 - 4104*K2**2 - 2060*K3**2 - 570*K4**2 + 4136
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1967']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16948', 'vk6.17191', 'vk6.20540', 'vk6.21940', 'vk6.23344', 'vk6.23639', 'vk6.27995', 'vk6.29461', 'vk6.35388', 'vk6.35809', 'vk6.39405', 'vk6.41598', 'vk6.42861', 'vk6.43140', 'vk6.45981', 'vk6.47657', 'vk6.55111', 'vk6.55372', 'vk6.57403', 'vk6.58578', 'vk6.59509', 'vk6.59809', 'vk6.62070', 'vk6.63053', 'vk6.64956', 'vk6.65164', 'vk6.66950', 'vk6.67810', 'vk6.68245', 'vk6.68388', 'vk6.69562', 'vk6.70258']
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 O1O2U3O4O5U1U6O3U5O6U4U2
R3 orbit {'O1O2U3O4O5U1U6O3U5O6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2U3O4O5U4U2O6U1O3U6U5
Gauss code of K* O1O2U3O4U2O5O6U1U6O3U5U4
Gauss code of -K* O1O2U3O4U5O3O6U4U2O5U1U6
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 1 0 0 1 0],[ 2 0 2 2 1 0 2],[-1 -2 0 0 -1 0 -1],[ 0 -2 0 0 0 1 0],[ 0 -1 1 0 0 1 -1],[-1 0 0 -1 -1 0 -1],[ 0 -2 1 0 1 1 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 0 0 -1 -1 -2],[-1 0 0 -1 -1 -1 0],[ 0 0 1 0 0 0 -2],[ 0 1 1 0 0 1 -2],[ 0 1 1 0 -1 0 -1],[ 2 2 0 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,0,0,1,1,2,1,1,1,0,0,0,2,-1,2,1]
Phi over symmetry [-2,0,0,0,1,1,0,0,1,1,3,0,-1,0,0,0,1,0,0,0,0]
Phi of -K [-2,0,0,0,1,1,0,0,1,1,3,0,-1,0,0,0,1,0,0,0,0]
Phi of K* [-1,-1,0,0,0,2,0,0,0,0,3,0,0,1,1,-1,0,1,0,0,0]
Phi of -K* [-2,0,0,0,1,1,1,2,2,0,2,-1,0,1,1,0,1,1,1,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+19t^4+38t^2+9
Outer characteristic polynomial t^7+25t^5+55t^3+19t
Flat arrow polynomial -6*K1**2 + 3*K2 + 4
2-strand cable arrow polynomial -64*K1**6 + 1248*K1**4*K2 - 3696*K1**4 + 32*K1**3*K2*K3 - 576*K1**3*K3 + 352*K1**2*K2**3 - 4448*K1**2*K2**2 - 608*K1**2*K2*K4 + 7896*K1**2*K2 - 112*K1**2*K3**2 - 4524*K1**2 - 576*K1*K2**2*K3 + 6056*K1*K2*K3 + 1080*K1*K3*K4 - 152*K2**4 + 688*K2**2*K4 - 4104*K2**2 - 2060*K3**2 - 570*K4**2 + 4136
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
Contact