| Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,1,1,1,-1,1,1,2,-1,1,0,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.2010'] |
| 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+18t^5+74t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2010'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 736*K1**4*K2 - 1936*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 + 1024*K1**2*K2**3 - 5504*K1**2*K2**2 - 64*K1**2*K2*K4 + 5912*K1**2*K2 - 16*K1**2*K3**2 - 2456*K1**2 - 768*K1*K2**2*K3 + 3800*K1*K2*K3 + 56*K1*K3*K4 - 1168*K2**4 + 912*K2**2*K4 - 1736*K2**2 - 656*K3**2 - 116*K4**2 + 2106 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2010'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17088', 'vk6.17330', 'vk6.20248', 'vk6.21553', 'vk6.23471', 'vk6.23808', 'vk6.27475', 'vk6.29070', 'vk6.35616', 'vk6.36062', 'vk6.38890', 'vk6.41090', 'vk6.42985', 'vk6.43296', 'vk6.45651', 'vk6.47384', 'vk6.55233', 'vk6.55484', 'vk6.57075', 'vk6.58226', 'vk6.59630', 'vk6.59975', 'vk6.61611', 'vk6.62785', 'vk6.65033', 'vk6.65233', 'vk6.66703', 'vk6.67560', 'vk6.68299', 'vk6.68448', 'vk6.69357', 'vk6.70099'] |
| 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 | O1O2U3O4U5O3U1U2O5O6U4U6 |
| R3 orbit | {'O1O2U3O4U5O3U1U2O5O6U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2U3U4O3O5U1U2O6U5O4U6 |
| Gauss code of K* | O1O2U3U4O5O4U1U2O6U5O3U6 |
| Gauss code of -K* | O1O2U3O4U5O3U1U2O6O5U6U4 |
| 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 -1 1 0 0 -1 1],[ 1 0 1 1 0 1 1],[-1 -1 0 -1 -1 0 1],[ 0 -1 1 0 1 -2 0],[ 0 0 1 -1 0 0 1],[ 1 -1 0 2 0 0 1],[-1 -1 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 -1 -1],[-1 0 1 -1 -1 0 -1],[-1 -1 0 0 -1 -1 -1],[ 0 1 0 0 1 -2 -1],[ 0 1 1 -1 0 0 0],[ 1 0 1 2 0 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,-1,1,1,0,1,0,1,1,1,-1,2,1,0,0,1] |
| Phi over symmetry | [-1,-1,0,0,1,1,-1,0,1,1,1,-1,1,1,2,-1,1,0,0,0,1] |
| Phi of -K | [-1,-1,0,0,1,1,-1,0,1,1,1,-1,1,1,2,-1,1,0,0,0,1] |
| Phi of K* | [-1,-1,0,0,1,1,-1,0,1,1,1,0,0,1,2,-1,1,1,0,-1,1] |
| Phi of -K* | [-1,-1,0,0,1,1,-1,0,2,0,1,0,1,1,1,-1,1,1,1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+14t^4+48t^2+4 |
| Outer characteristic polynomial | t^7+18t^5+74t^3+8t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 736*K1**4*K2 - 1936*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 + 1024*K1**2*K2**3 - 5504*K1**2*K2**2 - 64*K1**2*K2*K4 + 5912*K1**2*K2 - 16*K1**2*K3**2 - 2456*K1**2 - 768*K1*K2**2*K3 + 3800*K1*K2*K3 + 56*K1*K3*K4 - 1168*K2**4 + 912*K2**2*K4 - 1736*K2**2 - 656*K3**2 - 116*K4**2 + 2106 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}]] |
| If K is slice | True |