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

Flat knot 6.1430

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,2,2,1,0,0,1,0,1,1,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1430']
Arrow polynomial of the knot is: -10*K1**2 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.947', '6.1027', '6.1399', '6.1430', '6.1433', '6.1442', '6.1465', '6.1469', '6.1476', '6.1505', '6.1529', '6.1606', '6.1612', '6.1613', '6.1616', '6.1649', '6.1694', '6.1736', '6.1768', '6.1771', '6.1774', '6.1884', '6.1886', '6.1887', '6.1889', '6.1960', '6.1962']
Outer characteristic polynomial of the knot is: t^7+26t^5+34t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1430']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 384*K1**4*K2**2 + 3744*K1**4*K2 - 7248*K1**4 + 608*K1**3*K2*K3 - 1056*K1**3*K3 + 288*K1**2*K2**3 - 7840*K1**2*K2**2 - 160*K1**2*K2*K4 + 10896*K1**2*K2 - 624*K1**2*K3**2 - 2892*K1**2 - 256*K1*K2**2*K3 + 5744*K1*K2*K3 + 448*K1*K3*K4 - 200*K2**4 + 184*K2**2*K4 - 3472*K2**2 - 1076*K3**2 - 94*K4**2 + 3580
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1430']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4200', 'vk6.4281', 'vk6.5455', 'vk6.5568', 'vk6.7567', 'vk6.7657', 'vk6.9069', 'vk6.9150', 'vk6.11162', 'vk6.12248', 'vk6.12357', 'vk6.19382', 'vk6.19677', 'vk6.19775', 'vk6.26164', 'vk6.26212', 'vk6.26582', 'vk6.26657', 'vk6.30760', 'vk6.31965', 'vk6.38172', 'vk6.38196', 'vk6.44833', 'vk6.44941', 'vk6.48522', 'vk6.49217', 'vk6.49326', 'vk6.50312', 'vk6.52750', 'vk6.63592', 'vk6.66324', 'vk6.66352']
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 O1O2O3U2O4U3O5U1U4O6U5U6
R3 orbit {'O1O2O3U2O4U3O5U1U4O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U6U3O5U1O6U2
Gauss code of K* O1O2U3O4O3U1U5U6O5U2O6U4
Gauss code of -K* O1O2U1O3O4U2O5U3O6U5U6U4
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 1 1 1],[ 2 0 -1 1 2 2 1],[ 1 1 0 1 1 0 0],[ 0 -1 -1 0 1 1 0],[-1 -2 -1 -1 0 1 1],[-1 -2 0 -1 -1 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 -1 -1 -2],[-1 -1 0 1 -1 0 -2],[-1 -1 -1 0 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 1 1 0 0 1 0 1],[ 2 2 2 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,-1,1,1,2,-1,1,0,2,0,0,1,1,1,-1]
Phi over symmetry [-2,-1,0,1,1,1,-1,1,1,2,2,1,0,0,1,0,1,1,-1,-1,-1]
Phi of -K [-2,-1,0,1,1,1,2,1,1,1,2,0,1,2,2,0,0,1,-1,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,1,2,2,-1,0,2,1,0,1,1,0,1,2]
Phi of -K* [-2,-1,0,1,1,1,-1,1,1,2,2,1,0,0,1,0,1,1,-1,-1,-1]
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+18t^4+17t^2+1
Outer characteristic polynomial t^7+26t^5+34t^3+4t
Flat arrow polynomial -10*K1**2 + 5*K2 + 6
2-strand cable arrow polynomial -192*K1**6 - 384*K1**4*K2**2 + 3744*K1**4*K2 - 7248*K1**4 + 608*K1**3*K2*K3 - 1056*K1**3*K3 + 288*K1**2*K2**3 - 7840*K1**2*K2**2 - 160*K1**2*K2*K4 + 10896*K1**2*K2 - 624*K1**2*K3**2 - 2892*K1**2 - 256*K1*K2**2*K3 + 5744*K1*K2*K3 + 448*K1*K3*K4 - 200*K2**4 + 184*K2**2*K4 - 3472*K2**2 - 1076*K3**2 - 94*K4**2 + 3580
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
Contact