| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.782', '6.1236', '7.38129', '7.41463'] |
| 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+22t^5+31t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1236', '6.1954', '7.38129', '7.41463'] |
| 2-strand cable arrow polynomial of the knot is: -2432*K1**4*K2**2 + 3968*K1**4*K2 - 4864*K1**4 + 1792*K1**3*K2*K3 - 832*K1**3*K3 - 2112*K1**2*K2**4 + 5568*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 12000*K1**2*K2**2 - 1152*K1**2*K2*K4 + 7784*K1**2*K2 - 480*K1**2*K3**2 - 812*K1**2 + 2432*K1*K2**3*K3 - 2016*K1*K2**2*K3 - 480*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6128*K1*K2*K3 + 312*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3160*K2**4 - 592*K2**2*K3**2 - 48*K2**2*K4**2 + 1816*K2**2*K4 - 294*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 524*K3**2 - 82*K4**2 - 8*K5**2 - 2*K6**2 + 1720 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1236'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.45', 'vk6.90', 'vk6.185', 'vk6.244', 'vk6.284', 'vk6.663', 'vk6.670', 'vk6.1242', 'vk6.1331', 'vk6.1388', 'vk6.1430', 'vk6.1914', 'vk6.2365', 'vk6.2419', 'vk6.2619', 'vk6.2967', 'vk6.10086', 'vk6.10097', 'vk6.14590', 'vk6.15810', 'vk6.16213', 'vk6.17768', 'vk6.24272', 'vk6.29831', 'vk6.33398', 'vk6.33467', 'vk6.33540', 'vk6.36578', 'vk6.43690', 'vk6.53713', 'vk6.53772', 'vk6.63295'] |
| 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 | O1O2O3O4U3U4U5O6O5U1U2U6 |
| R3 orbit | {'O1O2O3O4U3U4U5O6O5U1U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U4O6O5U6U1U2 |
| Gauss code of K* | O1O2O3U4U3O5O6O4U5U6U1U2 |
| Gauss code of -K* | O1O2O3U4U1O4O5O6U5U6U2U3 |
| 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 1 1 1],[ 2 0 1 -1 1 2 1],[ 0 -1 0 -1 1 0 0],[ 1 1 1 0 1 1 0],[-1 -1 -1 -1 0 -1 0],[-1 -2 0 -1 1 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 -1 -1 -1],[ 0 0 0 1 0 -1 -1],[ 1 1 0 1 1 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,-1,0,1,2,0,0,0,1,1,1,1,1,1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,-1,-1] |
| Phi of -K | [-2,-1,0,1,1,1,2,1,1,2,2,0,1,1,2,1,0,1,-1,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,2,1,1,1,1,1,2,2,0,1,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+25z+27 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+25w^2z+27w |
| Inner characteristic polynomial | t^6+14t^4+14t^2+1 |
| Outer characteristic polynomial | t^7+22t^5+31t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -2432*K1**4*K2**2 + 3968*K1**4*K2 - 4864*K1**4 + 1792*K1**3*K2*K3 - 832*K1**3*K3 - 2112*K1**2*K2**4 + 5568*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 12000*K1**2*K2**2 - 1152*K1**2*K2*K4 + 7784*K1**2*K2 - 480*K1**2*K3**2 - 812*K1**2 + 2432*K1*K2**3*K3 - 2016*K1*K2**2*K3 - 480*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6128*K1*K2*K3 + 312*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3160*K2**4 - 592*K2**2*K3**2 - 48*K2**2*K4**2 + 1816*K2**2*K4 - 294*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 524*K3**2 - 82*K4**2 - 8*K5**2 - 2*K6**2 + 1720 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |