| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,2,1,2,1,1,0,-1,1,0,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1132'] |
| 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+33t^5+115t^3+19t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1132'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 128*K1**4*K2 - 2432*K1**4 + 384*K1**3*K2*K3 - 256*K1**3*K3 + 192*K1**2*K2**3 - 4832*K1**2*K2**2 - 192*K1**2*K2*K4 + 7360*K1**2*K2 - 1216*K1**2*K3**2 - 4408*K1**2 - 384*K1*K2**2*K3 + 6592*K1*K2*K3 + 1184*K1*K3*K4 - 128*K2**4 + 272*K2**2*K4 - 3600*K2**2 - 1992*K3**2 - 312*K4**2 + 3766 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1132'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71351', 'vk6.71401', 'vk6.71862', 'vk6.71923', 'vk6.74329', 'vk6.74976', 'vk6.76541', 'vk6.76949', 'vk6.77005', 'vk6.77064', 'vk6.77390', 'vk6.79374', 'vk6.79801', 'vk6.80832', 'vk6.81280', 'vk6.81477', 'vk6.83838', 'vk6.87057', 'vk6.88051', 'vk6.89558'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is r. |
| 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 | O1O2O3O4U5U3O6U4U1O5U2U6 |
| R3 orbit | {'O1O2O3O4U5U3O6U4U1O5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | Same |
| Gauss code of K* | O1O2U3O4O5U1O6O3U5U6U2U4 |
| Gauss code of -K* | O1O2U3O4O5U1O6O3U5U6U2U4 |
| Diagrammatic symmetry type | r |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -1 0 0 1 -2 2],[ 1 0 0 -1 1 -1 2],[ 0 0 0 0 2 -2 1],[ 0 1 0 0 1 -1 0],[-1 -1 -2 -1 0 -1 0],[ 2 1 2 1 1 0 2],[-2 -2 -1 0 0 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -2 -2],[-1 0 0 -1 -2 -1 -1],[ 0 0 1 0 0 1 -1],[ 0 1 2 0 0 0 -2],[ 1 2 1 -1 0 0 -1],[ 2 2 1 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,1,2,2,1,2,1,1,0,-1,1,0,2,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,2,2,1,2,1,1,0,-1,1,0,2,1] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,2,2,1,2,1,1,0,-1,1,0,2,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,2,1,2,-1,0,1,2,0,1,0,2,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,1,2,-1,0,1,2,0,1,0,2,1,0] |
| Symmetry type of based matrix | r |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+23t^4+63t^2+9 |
| Outer characteristic polynomial | t^7+33t^5+115t^3+19t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 128*K1**4*K2 - 2432*K1**4 + 384*K1**3*K2*K3 - 256*K1**3*K3 + 192*K1**2*K2**3 - 4832*K1**2*K2**2 - 192*K1**2*K2*K4 + 7360*K1**2*K2 - 1216*K1**2*K3**2 - 4408*K1**2 - 384*K1*K2**2*K3 + 6592*K1*K2*K3 + 1184*K1*K3*K4 - 128*K2**4 + 272*K2**2*K4 - 3600*K2**2 - 1992*K3**2 - 312*K4**2 + 3766 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]] |
| If K is slice | False |