| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,1,1,0,1,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1232', '7.41172'] |
| 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+33t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1232', '7.41172'] |
| 2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1664*K1**4*K2 - 4352*K1**4 + 1920*K1**3*K2*K3 - 1024*K1**3*K3 - 832*K1**2*K2**4 + 2560*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 9696*K1**2*K2**2 - 1056*K1**2*K2*K4 + 8960*K1**2*K2 - 1312*K1**2*K3**2 - 2452*K1**2 + 2400*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 576*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 8016*K1*K2*K3 + 1144*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2296*K2**4 - 32*K2**3*K6 - 1584*K2**2*K3**2 - 112*K2**2*K4**2 + 1656*K2**2*K4 - 1902*K2**2 - 96*K2*K3**2*K4 + 808*K2*K3*K5 + 88*K2*K4*K6 + 24*K3**2*K6 - 1492*K3**2 - 266*K4**2 - 56*K5**2 - 18*K6**2 + 2816 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1232'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.337', 'vk6.378', 'vk6.447', 'vk6.734', 'vk6.789', 'vk6.898', 'vk6.1469', 'vk6.1528', 'vk6.1603', 'vk6.1967', 'vk6.2008', 'vk6.2084', 'vk6.2487', 'vk6.2746', 'vk6.3006', 'vk6.3129', 'vk6.3776', 'vk6.3969', 'vk6.7168', 'vk6.7345', 'vk6.18793', 'vk6.19858', 'vk6.24923', 'vk6.25386', 'vk6.25926', 'vk6.26303', 'vk6.26746', 'vk6.38005', 'vk6.38060', 'vk6.45046', 'vk6.50090', 'vk6.60756'] |
| 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 | O1O2O3O4U3U4U1O5O6U5U2U6 |
| R3 orbit | {'O1O2O3O4U3U4U1O5O6U5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U6O5O6U4U1U2 |
| Gauss code of K* | O1O2O3U4U5O4O6O5U3U6U1U2 |
| Gauss code of -K* | O1O2O3U1U3O4O5O6U5U6U2U4 |
| 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 0 -1 1 -1 2],[ 1 0 1 -1 1 0 1],[ 0 -1 0 -1 1 0 2],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -2 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -2 0 -1 -1],[-1 0 0 -1 -1 0 -1],[ 0 2 1 0 -1 0 -1],[ 1 0 1 1 0 0 1],[ 1 1 0 0 0 0 0],[ 1 1 1 1 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,1,1,0,1,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,1,1,0,1,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,3,0,0,1,2,1,2,2,0,0,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,0,2,2,3,0,1,2,1,0,1,0,0,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,1,1,0,1,1,0,0,0,1,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+12t^4+14t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+33t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -896*K1**4*K2**2 + 1664*K1**4*K2 - 4352*K1**4 + 1920*K1**3*K2*K3 - 1024*K1**3*K3 - 832*K1**2*K2**4 + 2560*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 9696*K1**2*K2**2 - 1056*K1**2*K2*K4 + 8960*K1**2*K2 - 1312*K1**2*K3**2 - 2452*K1**2 + 2400*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 576*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 8016*K1*K2*K3 + 1144*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2296*K2**4 - 32*K2**3*K6 - 1584*K2**2*K3**2 - 112*K2**2*K4**2 + 1656*K2**2*K4 - 1902*K2**2 - 96*K2*K3**2*K4 + 808*K2*K3*K5 + 88*K2*K4*K6 + 24*K3**2*K6 - 1492*K3**2 - 266*K4**2 - 56*K5**2 - 18*K6**2 + 2816 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |