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

Flat knot 6.1505

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,3,-1,0,0,0,0,0,0,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1505']
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+23t^5+31t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1505']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 192*K1**4*K2**2 + 2176*K1**4*K2 - 6816*K1**4 + 608*K1**3*K2*K3 - 1856*K1**3*K3 - 5520*K1**2*K2**2 - 736*K1**2*K2*K4 + 13360*K1**2*K2 - 480*K1**2*K3**2 - 6516*K1**2 - 96*K1*K2**2*K3 + 7456*K1*K2*K3 + 872*K1*K3*K4 - 104*K2**4 + 408*K2**2*K4 - 5568*K2**2 - 2132*K3**2 - 350*K4**2 + 5612
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1505']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3661', 'vk6.3758', 'vk6.3947', 'vk6.4044', 'vk6.4500', 'vk6.4595', 'vk6.5886', 'vk6.6013', 'vk6.7146', 'vk6.7319', 'vk6.7412', 'vk6.7931', 'vk6.8050', 'vk6.9365', 'vk6.17906', 'vk6.18003', 'vk6.18751', 'vk6.24441', 'vk6.24876', 'vk6.25337', 'vk6.37490', 'vk6.43868', 'vk6.44223', 'vk6.44526', 'vk6.48301', 'vk6.48366', 'vk6.50084', 'vk6.50194', 'vk6.50576', 'vk6.50639', 'vk6.55867', 'vk6.60715']
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 O1O2O3U1O4U5U4O6O5U3U6U2
R3 orbit {'O1O2O3U1O4U5U4O6O5U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U1O5O4U6U5O6U3
Gauss code of K* O1O2O3U4U3U1O4U5O6O5U2U6
Gauss code of -K* O1O2O3U4U2O5O4U5O6U3U1U6
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 0 0],[ 2 0 2 1 0 2 1],[-1 -2 0 -1 1 -1 0],[ 0 -1 1 0 1 -1 0],[-1 0 -1 -1 0 -1 -1],[ 0 -2 1 1 1 0 0],[ 0 -1 0 0 1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 1 0 -1 -1 -2],[-1 -1 0 -1 -1 -1 0],[ 0 0 1 0 0 0 -1],[ 0 1 1 0 0 1 -2],[ 0 1 1 0 -1 0 -1],[ 2 2 0 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,-1,0,1,1,2,1,1,1,0,0,0,1,-1,2,1]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,1,3,-1,0,0,0,0,0,0,1,0,-1]
Phi of -K [-2,0,0,0,1,1,0,1,1,1,3,-1,0,0,0,0,0,0,1,0,-1]
Phi of K* [-1,-1,0,0,0,2,-1,0,0,0,3,0,0,1,1,-1,0,1,0,0,1]
Phi of -K* [-2,0,0,0,1,1,1,1,2,0,2,0,-1,1,1,0,1,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+17t^4+20t^2+4
Outer characteristic polynomial t^7+23t^5+31t^3+8t
Flat arrow polynomial -10*K1**2 + 5*K2 + 6
2-strand cable arrow polynomial -128*K1**6 - 192*K1**4*K2**2 + 2176*K1**4*K2 - 6816*K1**4 + 608*K1**3*K2*K3 - 1856*K1**3*K3 - 5520*K1**2*K2**2 - 736*K1**2*K2*K4 + 13360*K1**2*K2 - 480*K1**2*K3**2 - 6516*K1**2 - 96*K1*K2**2*K3 + 7456*K1*K2*K3 + 872*K1*K3*K4 - 104*K2**4 + 408*K2**2*K4 - 5568*K2**2 - 2132*K3**2 - 350*K4**2 + 5612
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
Contact