| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,3,-1,1,0,1,0,0,0,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1792'] |
| 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+25t^5+37t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1792'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 64*K1**4*K2 - 1536*K1**4 + 224*K1**3*K2*K3 - 320*K1**3*K3 + 96*K1**2*K2**3 - 2816*K1**2*K2**2 - 352*K1**2*K2*K4 + 5480*K1**2*K2 - 320*K1**2*K3**2 - 3816*K1**2 - 192*K1*K2**2*K3 + 4376*K1*K2*K3 + 592*K1*K3*K4 - 96*K2**4 + 352*K2**2*K4 - 2960*K2**2 - 1416*K3**2 - 268*K4**2 + 2970 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1792'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71371', 'vk6.71432', 'vk6.71893', 'vk6.71954', 'vk6.72449', 'vk6.72612', 'vk6.72729', 'vk6.72810', 'vk6.72875', 'vk6.73040', 'vk6.74223', 'vk6.74368', 'vk6.74426', 'vk6.74852', 'vk6.75040', 'vk6.76615', 'vk6.76903', 'vk6.77036', 'vk6.77406', 'vk6.77749', 'vk6.77802', 'vk6.79269', 'vk6.79414', 'vk6.79743', 'vk6.79830', 'vk6.79883', 'vk6.80864', 'vk6.80912', 'vk6.81382', 'vk6.85523', 'vk6.87202', 'vk6.89267'] |
| 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 | O1O2O3U4U2O5O6U1U5O4U6U3 |
| R3 orbit | {'O1O2O3U4U2O5O6U1U5O4U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U6U3O4O6U2U5 |
| Gauss code of K* | O1O2U3O4O5U1U6U5O3O6U2U4 |
| Gauss code of -K* | O1O2U3O4O5U2U4O6O3U1U6U5 |
| 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 0 2 -1 0 1],[ 2 0 0 3 0 1 1],[ 0 0 0 0 0 0 -1],[-2 -3 0 0 -1 -1 0],[ 1 0 0 1 0 0 0],[ 0 -1 0 1 0 0 1],[-1 -1 1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -1 -3],[-1 0 0 1 -1 0 -1],[ 0 0 -1 0 0 0 0],[ 0 1 1 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 2 3 1 0 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,1,1,3,-1,1,0,1,0,0,0,0,1,0] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,1,3,-1,1,0,1,0,0,0,0,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,1,1,2,2,1,1,1,2,2,0,0,1,2,2,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,2,2,1,0,2,2,2,0,1,1,1,2,1] |
| Phi of -K* | [-2,-1,0,0,1,2,0,0,1,1,3,0,0,0,1,0,-1,0,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 2z^2+19z+31 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+19w^2z+31w |
| Inner characteristic polynomial | t^6+15t^4+19t^2+1 |
| Outer characteristic polynomial | t^7+25t^5+37t^3+5t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 64*K1**4*K2 - 1536*K1**4 + 224*K1**3*K2*K3 - 320*K1**3*K3 + 96*K1**2*K2**3 - 2816*K1**2*K2**2 - 352*K1**2*K2*K4 + 5480*K1**2*K2 - 320*K1**2*K3**2 - 3816*K1**2 - 192*K1*K2**2*K3 + 4376*K1*K2*K3 + 592*K1*K3*K4 - 96*K2**4 + 352*K2**2*K4 - 2960*K2**2 - 1416*K3**2 - 268*K4**2 + 2970 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | True |