| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,2,3,0,1,1,1,0,0,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1730'] |
| Arrow polynomial of the knot is: -6*K1**2 + 3*K2 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.689', '6.691', '6.752', '6.754', '6.1106', '6.1116', '6.1126', '6.1335', '6.1379', '6.1386', '6.1409', '6.1415', '6.1417', '6.1418', '6.1421', '6.1422', '6.1428', '6.1431', '6.1432', '6.1435', '6.1443', '6.1445', '6.1446', '6.1447', '6.1454', '6.1455', '6.1460', '6.1462', '6.1464', '6.1466', '6.1472', '6.1474', '6.1475', '6.1501', '6.1516', '6.1518', '6.1566', '6.1570', '6.1590', '6.1599', '6.1602', '6.1603', '6.1604', '6.1605', '6.1614', '6.1615', '6.1625', '6.1628', '6.1730', '6.1780', '6.1883', '6.1885', '6.1888', '6.1890', '6.1941', '6.1943', '6.1945', '6.1948', '6.1961', '6.1963', '6.1966', '6.1967', '6.1971'] |
| Outer characteristic polynomial of the knot is: t^7+28t^5+28t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1730'] |
| 2-strand cable arrow polynomial of the knot is: 320*K1**4*K2 - 2976*K1**4 + 64*K1**3*K2*K3 - 384*K1**3*K3 - 2544*K1**2*K2**2 - 64*K1**2*K2*K4 + 6024*K1**2*K2 - 192*K1**2*K3**2 - 2564*K1**2 - 224*K1*K2**2*K3 + 3048*K1*K2*K3 + 328*K1*K3*K4 - 120*K2**4 + 240*K2**2*K4 - 2352*K2**2 - 836*K3**2 - 142*K4**2 + 2372 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1730'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4364', 'vk6.4395', 'vk6.5686', 'vk6.5717', 'vk6.7747', 'vk6.7778', 'vk6.9229', 'vk6.9260', 'vk6.10496', 'vk6.10554', 'vk6.10651', 'vk6.10719', 'vk6.10750', 'vk6.10836', 'vk6.14604', 'vk6.15304', 'vk6.15431', 'vk6.16227', 'vk6.17967', 'vk6.24405', 'vk6.30183', 'vk6.30241', 'vk6.30338', 'vk6.30463', 'vk6.33950', 'vk6.34351', 'vk6.34407', 'vk6.43840', 'vk6.50449', 'vk6.50480', 'vk6.54192', 'vk6.63439'] |
| 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 | O1O2O3U2U4O5O4U1U5O6U3U6 |
| R3 orbit | {'O1O2O3U2U4O5O4U1U5O6U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1O4U5U3O6O5U6U2 |
| Gauss code of K* | O1O2U3O4O3U1U5U4O5O6U2U6 |
| Gauss code of -K* | O1O2U1O3O4U5U3O5O6U2U6U4 |
| 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 1 1 0 1],[ 2 0 0 3 2 0 1],[ 1 0 0 1 1 0 1],[-1 -3 -1 0 0 -1 1],[-1 -2 -1 0 0 0 1],[ 0 0 0 1 0 0 0],[-1 -1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 -1 0 -1 -1],[-1 0 1 0 -1 -1 -3],[ 0 0 0 1 0 0 0],[ 1 1 1 1 0 0 0],[ 2 2 1 3 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,2,1,0,1,1,1,1,3,0,0,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,2,3,0,1,1,1,0,0,1,-1,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,2,0,1,2,1,1,1,1,0,1,1,0,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,1,1,2,0,0,1,0,1,1,1,1,2,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,0,1,2,3,0,1,1,1,0,0,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+20t^4+15t^2+1 |
| Outer characteristic polynomial | t^7+28t^5+28t^3+5t |
| Flat arrow polynomial | -6*K1**2 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | 320*K1**4*K2 - 2976*K1**4 + 64*K1**3*K2*K3 - 384*K1**3*K3 - 2544*K1**2*K2**2 - 64*K1**2*K2*K4 + 6024*K1**2*K2 - 192*K1**2*K3**2 - 2564*K1**2 - 224*K1*K2**2*K3 + 3048*K1*K2*K3 + 328*K1*K3*K4 - 120*K2**4 + 240*K2**2*K4 - 2352*K2**2 - 836*K3**2 - 142*K4**2 + 2372 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {1, 4}, {3}]] |
| If K is slice | False |