| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,0,0,0,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1844', '7.41196'] |
| 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+20t^5+23t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1844', '7.35146'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 320*K1**4*K2**2 + 1344*K1**4*K2 - 2768*K1**4 + 512*K1**3*K2*K3 - 320*K1**3*K3 - 192*K1**2*K2**4 + 1024*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 4432*K1**2*K2**2 - 352*K1**2*K2*K4 + 4192*K1**2*K2 - 400*K1**2*K3**2 - 48*K1**2*K4**2 - 364*K1**2 + 320*K1*K2**3*K3 - 576*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2784*K1*K2*K3 + 304*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 776*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 536*K2**2*K4 - 710*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 78*K4**2 - 16*K5**2 - 2*K6**2 + 1028 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1844'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.65', 'vk6.120', 'vk6.219', 'vk6.266', 'vk6.296', 'vk6.682', 'vk6.711', 'vk6.756', 'vk6.1225', 'vk6.1272', 'vk6.1365', 'vk6.1412', 'vk6.1505', 'vk6.1565', 'vk6.1934', 'vk6.2042', 'vk6.2437', 'vk6.2484', 'vk6.2643', 'vk6.2987', 'vk6.5754', 'vk6.5785', 'vk6.7823', 'vk6.7854', 'vk6.10258', 'vk6.10401', 'vk6.13296', 'vk6.13327', 'vk6.14780', 'vk6.14801', 'vk6.15940', 'vk6.15959', 'vk6.18052', 'vk6.24496', 'vk6.25847', 'vk6.33053', 'vk6.37400', 'vk6.37954', 'vk6.38019', 'vk6.44854'] |
| 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 | O1O2O3U2U3O4U5O6O5U1U4U6 |
| R3 orbit | {'O1O2O3U2U3O4U5O6O5U1U4U6', 'O1O2O3U2U3U4O5O4O6U1U6U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U5U3O6O4U6O5U1U2 |
| Gauss code of K* | O1O2O3U1U4U5O4O5U2O6U3U6 |
| Gauss code of -K* | O1O2O3U4U1O4U2O5O6U5U6U3 |
| 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 0 1 1],[ 2 0 -1 1 1 2 1],[ 1 1 0 1 0 1 0],[-1 -1 -1 0 0 -1 0],[ 0 -1 0 0 0 0 0],[-1 -2 -1 1 0 0 1],[-1 -1 0 0 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 0 0 0 -1],[-1 -1 0 0 0 -1 -1],[ 0 0 0 0 0 0 -1],[ 1 1 0 1 0 0 1],[ 2 2 1 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,0,0,1,0,1,1,0,1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,0,0,0,0,-1,-1] |
| Phi of -K | [-2,-1,0,1,1,1,2,1,1,2,2,1,1,1,2,1,1,1,-1,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,2,1,1,1,1,1,2,2,1,1,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,0,0,0,0,-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+12t^4+10t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+23t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -256*K1**6 - 320*K1**4*K2**2 + 1344*K1**4*K2 - 2768*K1**4 + 512*K1**3*K2*K3 - 320*K1**3*K3 - 192*K1**2*K2**4 + 1024*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 4432*K1**2*K2**2 - 352*K1**2*K2*K4 + 4192*K1**2*K2 - 400*K1**2*K3**2 - 48*K1**2*K4**2 - 364*K1**2 + 320*K1*K2**3*K3 - 576*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2784*K1*K2*K3 + 304*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 776*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 536*K2**2*K4 - 710*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 78*K4**2 - 16*K5**2 - 2*K6**2 + 1028 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}]] |
| If K is slice | False |