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

Flat knot 6.1536

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,1,1,1,0,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1536', '7.41386']
Arrow polynomial of the knot is: -8*K1**2 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.668', '6.711', '6.777', '6.803', '6.828', '6.1015', '6.1032', '6.1055', '6.1082', '6.1132', '6.1264', '6.1288', '6.1333', '6.1391', '6.1395', '6.1396', '6.1400', '6.1404', '6.1405', '6.1419', '6.1471', '6.1473', '6.1536', '6.1563', '6.1611', '6.1618', '6.1623', '6.1627', '6.1629', '6.1631', '6.1695', '6.1700', '6.1731', '6.1740', '6.1767', '6.1773', '6.1790', '6.1792', '6.1796', '6.1848', '6.1899', '6.1901', '6.1937', '6.1954', '6.1955', '6.1958', '6.1964', '6.1975', '6.1997', '6.1998', '6.1999', '6.2002', '6.2003', '6.2004', '6.2005', '6.2007', '6.2008', '6.2009', '6.2010', '6.2011', '6.2013', '6.2018', '6.2019', '6.2021', '6.2034', '6.2039', '6.2043', '6.2046', '6.2050', '6.2051', '6.2057', '6.2063']
Outer characteristic polynomial of the knot is: t^7+16t^5+12t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1536']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 256*K1**4*K2**2 + 896*K1**4*K2 - 1952*K1**4 + 224*K1**3*K2*K3 - 1280*K1**2*K2**2 + 2624*K1**2*K2 - 64*K1**2*K3**2 - 428*K1**2 + 864*K1*K2*K3 + 8*K1*K3*K4 - 64*K2**4 + 64*K2**2*K4 - 864*K2**2 - 188*K3**2 - 16*K4**2 + 878
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1536']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4141', 'vk6.4174', 'vk6.5379', 'vk6.5412', 'vk6.5475', 'vk6.5588', 'vk6.7501', 'vk6.7669', 'vk6.9002', 'vk6.9035', 'vk6.11180', 'vk6.12262', 'vk6.12371', 'vk6.12443', 'vk6.12474', 'vk6.13361', 'vk6.13580', 'vk6.13613', 'vk6.14258', 'vk6.14707', 'vk6.14731', 'vk6.15187', 'vk6.15861', 'vk6.15885', 'vk6.26194', 'vk6.26639', 'vk6.30856', 'vk6.30887', 'vk6.32040', 'vk6.32071', 'vk6.33085', 'vk6.33118', 'vk6.38158', 'vk6.38182', 'vk6.44815', 'vk6.44923', 'vk6.49233', 'vk6.49346', 'vk6.52760', 'vk6.53533']
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 O1O2O3U2O4U5U4O6O5U3U1U6
R3 orbit {'O1O2U1O3O4U5U4O6O5U2U3U6', 'O1O2O3U2O4U5U4O6O5U3U1U6'}
R3 orbit length 2
Gauss code of -K O1O2O3U4U3U1O5O4U6U5O6U2
Gauss code of K* O1O2O3U2U4U1O4U5O6O5U3U6
Gauss code of -K* O1O2O3U4U1O5O4U5O6U3U6U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -1 0 1 0 1],[ 1 0 -1 1 1 0 1],[ 1 1 0 1 0 1 1],[ 0 -1 -1 0 1 -1 0],[-1 -1 0 -1 0 -1 -1],[ 0 0 -1 1 1 0 1],[-1 -1 -1 0 1 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 -1 0 -1],[ 0 0 1 0 -1 -1 -1],[ 0 1 1 1 0 -1 0],[ 1 1 0 1 1 0 1],[ 1 1 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,0,1,1,1,1,1,0,-1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,1,1,1,0,0,0,-1]
Phi of -K [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,1,1,1,0,0,0,-1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,1,1,1,0,0,0,-1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,0,1,1,1,1,1,0,-1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 12z+25
Enhanced Jones-Krushkal polynomial 12w^2z+25w
Inner characteristic polynomial t^6+12t^4
Outer characteristic polynomial t^7+16t^5+12t^3
Flat arrow polynomial -8*K1**2 + 4*K2 + 5
2-strand cable arrow polynomial -448*K1**6 - 256*K1**4*K2**2 + 896*K1**4*K2 - 1952*K1**4 + 224*K1**3*K2*K3 - 1280*K1**2*K2**2 + 2624*K1**2*K2 - 64*K1**2*K3**2 - 428*K1**2 + 864*K1*K2*K3 + 8*K1*K3*K4 - 64*K2**4 + 64*K2**2*K4 - 864*K2**2 - 188*K3**2 - 16*K4**2 + 878
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice True
Contact