| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,1,2,2,-1,1,0,1,0,-1,-1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1780'] |
| 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+29t^5+111t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1780'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 128*K1**4*K2 - 976*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 + 256*K1**2*K2**3 - 2176*K1**2*K2**2 - 64*K1**2*K2*K4 + 3472*K1**2*K2 - 144*K1**2*K3**2 - 1888*K1**2 - 224*K1*K2**2*K3 + 2048*K1*K2*K3 + 104*K1*K3*K4 - 152*K2**4 + 152*K2**2*K4 - 1368*K2**2 - 472*K3**2 - 38*K4**2 + 1404 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1780'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73271', 'vk6.73414', 'vk6.73705', 'vk6.73822', 'vk6.74546', 'vk6.74812', 'vk6.75179', 'vk6.75630', 'vk6.76024', 'vk6.76367', 'vk6.76760', 'vk6.76875', 'vk6.78140', 'vk6.78613', 'vk6.78991', 'vk6.79227', 'vk6.79546', 'vk6.79700', 'vk6.79973', 'vk6.80251', 'vk6.80513', 'vk6.80708', 'vk6.80983', 'vk6.81070', 'vk6.81615', 'vk6.81793', 'vk6.82154', 'vk6.82171', 'vk6.82649', 'vk6.84033', 'vk6.84064', 'vk6.84223', 'vk6.84599', 'vk6.84931', 'vk6.85943', 'vk6.86733', 'vk6.87651', 'vk6.87766', 'vk6.88219', 'vk6.89978'] |
| 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 | O1O2O3U4U5O6O4U1U2O5U3U6 |
| R3 orbit | {'O1O2U3O4U5O3O6U1U2O5U6U4', 'O1O2O3U4U5O6O4U1U2O5U3U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U1O5U2U3O6O4U5U6 |
| Gauss code of K* | O1O2U3O4O5U1U2U4O6O3U5U6 |
| Gauss code of -K* | O1O2U3O4O5U6U1O3O6U2U4U5 |
| 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 1 0 0 1],[ 2 0 1 1 2 2 1],[ 0 -1 0 0 0 1 0],[-1 -1 0 0 -2 0 -1],[ 0 -2 0 2 0 -1 2],[ 0 -2 -1 0 1 0 1],[-1 -1 0 1 -2 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 1 0 -1 -2 -1],[-1 -1 0 0 0 -2 -1],[ 0 0 0 0 1 0 -1],[ 0 1 0 -1 0 1 -2],[ 0 2 2 0 -1 0 -2],[ 2 1 1 1 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,-1,0,1,2,1,0,0,2,1,-1,0,1,-1,2,2] |
| Phi over symmetry | [-2,0,0,0,1,1,0,0,1,2,2,-1,1,0,1,0,-1,-1,1,1,-1] |
| Phi of -K | [-2,0,0,0,1,1,0,0,1,2,2,-1,1,0,1,0,-1,-1,1,1,-1] |
| Phi of K* | [-1,-1,0,0,0,2,-1,-1,1,1,2,-1,0,1,2,-1,0,0,-1,0,1] |
| Phi of -K* | [-2,0,0,0,1,1,1,2,2,1,1,0,1,0,0,-1,2,2,0,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+23t^4+80t^2+1 |
| Outer characteristic polynomial | t^7+29t^5+111t^3+4t |
| Flat arrow polynomial | -6*K1**2 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 128*K1**4*K2 - 976*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 + 256*K1**2*K2**3 - 2176*K1**2*K2**2 - 64*K1**2*K2*K4 + 3472*K1**2*K2 - 144*K1**2*K3**2 - 1888*K1**2 - 224*K1*K2**2*K3 + 2048*K1*K2*K3 + 104*K1*K3*K4 - 152*K2**4 + 152*K2**2*K4 - 1368*K2**2 - 472*K3**2 - 38*K4**2 + 1404 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {1, 5}, {3, 4}, {2}]] |
| If K is slice | False |