| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,2,2,0,1,1,1,1,1,1,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1029'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+26t^5+42t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1029'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 256*K1**4*K2**2 + 2080*K1**4*K2 - 4048*K1**4 - 256*K1**3*K2**2*K3 + 448*K1**3*K2*K3 - 1536*K1**3*K3 + 1376*K1**2*K2**3 - 6160*K1**2*K2**2 + 288*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 9040*K1**2*K2 - 592*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 5100*K1**2 + 480*K1*K2**3*K3 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 7336*K1*K2*K3 + 1360*K1*K3*K4 + 120*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 920*K2**4 - 528*K2**2*K3**2 - 48*K2**2*K4**2 + 1040*K2**2*K4 - 3870*K2**2 + 328*K2*K3*K5 + 16*K2*K4*K6 - 2168*K3**2 - 654*K4**2 - 100*K5**2 - 2*K6**2 + 4404 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1029'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4066', 'vk6.4097', 'vk6.5304', 'vk6.5335', 'vk6.7438', 'vk6.7465', 'vk6.8935', 'vk6.8966', 'vk6.10106', 'vk6.10273', 'vk6.10296', 'vk6.14560', 'vk6.15295', 'vk6.15421', 'vk6.15784', 'vk6.16199', 'vk6.29856', 'vk6.29887', 'vk6.33929', 'vk6.34008', 'vk6.34227', 'vk6.34392', 'vk6.48452', 'vk6.49151', 'vk6.50206', 'vk6.50231', 'vk6.51582', 'vk6.53972', 'vk6.54033', 'vk6.54184', 'vk6.54477', 'vk6.63303'] |
| 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 | O1O2O3O4U3U5O6U4O5U1U6U2 |
| R3 orbit | {'O1O2O3O4U3U5O6U4O5U1U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U5U4O6U1O5U6U2 |
| Gauss code of K* | O1O2O3U1U3U4U5O4O6U2O5U6 |
| Gauss code of -K* | O1O2O3U4O5U2O4O6U5U6U1U3 |
| 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 2 -1 2 1 1],[-1 -2 0 -1 1 -1 0],[ 1 1 1 0 1 0 1],[-1 -2 -1 -1 0 -1 0],[ 0 -1 1 0 1 0 1],[-1 -1 0 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -2],[-1 -1 0 0 -1 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 1 1 0 0 -1],[ 1 1 1 1 0 0 1],[ 2 2 2 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,1,2,1,1,1,0,1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,2,2,0,1,1,1,1,1,1,0,0,-1] |
| Phi of -K | [-2,-1,0,1,1,1,2,1,1,1,2,1,1,1,1,0,0,0,-1,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,1,0,0,1,1,0,1,2,1,1,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,1,1,2,2,0,1,1,1,1,1,1,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+21z+35 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2-2w^3z+23w^2z+35w |
| Inner characteristic polynomial | t^6+18t^4+21t^2+4 |
| Outer characteristic polynomial | t^7+26t^5+42t^3+13t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 - 256*K1**4*K2**2 + 2080*K1**4*K2 - 4048*K1**4 - 256*K1**3*K2**2*K3 + 448*K1**3*K2*K3 - 1536*K1**3*K3 + 1376*K1**2*K2**3 - 6160*K1**2*K2**2 + 288*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 9040*K1**2*K2 - 592*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 5100*K1**2 + 480*K1*K2**3*K3 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 7336*K1*K2*K3 + 1360*K1*K3*K4 + 120*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 920*K2**4 - 528*K2**2*K3**2 - 48*K2**2*K4**2 + 1040*K2**2*K4 - 3870*K2**2 + 328*K2*K3*K5 + 16*K2*K4*K6 - 2168*K3**2 - 654*K4**2 - 100*K5**2 - 2*K6**2 + 4404 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |