| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,1,0,0,1,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1826', '7.42381'] |
| 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+18t^5+45t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1826'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 2624*K1**4*K2 - 6128*K1**4 + 1408*K1**3*K2*K3 - 1056*K1**3*K3 - 448*K1**2*K2**4 + 2592*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 11120*K1**2*K2**2 - 1344*K1**2*K2*K4 + 10496*K1**2*K2 - 1040*K1**2*K3**2 - 144*K1**2*K4**2 - 2156*K1**2 + 1600*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1632*K1*K2**2*K3 - 544*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8328*K1*K2*K3 - 96*K1*K2*K4*K5 + 1216*K1*K3*K4 + 208*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 2440*K2**4 - 32*K2**3*K6 - 1120*K2**2*K3**2 - 112*K2**2*K4**2 + 1752*K2**2*K4 - 2030*K2**2 + 592*K2*K3*K5 + 88*K2*K4*K6 - 1380*K3**2 - 358*K4**2 - 80*K5**2 - 18*K6**2 + 3084 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1826'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.504', 'vk6.597', 'vk6.622', 'vk6.1004', 'vk6.1103', 'vk6.1129', 'vk6.1663', 'vk6.1839', 'vk6.2178', 'vk6.2181', 'vk6.2287', 'vk6.2310', 'vk6.2783', 'vk6.2886', 'vk6.3066', 'vk6.3194', 'vk6.5248', 'vk6.6505', 'vk6.8885', 'vk6.9802', 'vk6.20815', 'vk6.21048', 'vk6.22212', 'vk6.22472', 'vk6.28497', 'vk6.29778', 'vk6.39875', 'vk6.40273', 'vk6.46429', 'vk6.46929', 'vk6.49120', 'vk6.58827'] |
| 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 | O1O2O3U1U3O4U2O5O6U5U6U4 |
| R3 orbit | {'O1O2O3U1U3O4U2O5O6U5U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U6O5O6U2O4U1U3 |
| Gauss code of K* | O1O2O3U4U5U6O4O6U3O5U1U2 |
| Gauss code of -K* | O1O2O3U2U3O4U1O5O6U5U4U6 |
| 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 2 1 1 0 0],[ 0 -2 0 0 1 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 -1 1],[ 1 0 0 0 1 0 1],[-1 0 0 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 0 0 -1 0],[-1 0 0 0 0 0 -1],[ 0 1 0 0 0 0 -2],[ 1 1 1 0 0 0 0],[ 2 1 0 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,1,0,0,1,0,0,0,1,0,2,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,1,0,0,1,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,0,2,2,3,1,1,2,1,0,1,1,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,3,0,0,1,2,1,2,2,1,0,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,1,0,0,1,0,-1,0] |
| 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+10t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+18t^5+45t^3+8t |
| 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 - 576*K1**4*K2**2 + 2624*K1**4*K2 - 6128*K1**4 + 1408*K1**3*K2*K3 - 1056*K1**3*K3 - 448*K1**2*K2**4 + 2592*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 11120*K1**2*K2**2 - 1344*K1**2*K2*K4 + 10496*K1**2*K2 - 1040*K1**2*K3**2 - 144*K1**2*K4**2 - 2156*K1**2 + 1600*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1632*K1*K2**2*K3 - 544*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8328*K1*K2*K3 - 96*K1*K2*K4*K5 + 1216*K1*K3*K4 + 208*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 2440*K2**4 - 32*K2**3*K6 - 1120*K2**2*K3**2 - 112*K2**2*K4**2 + 1752*K2**2*K4 - 2030*K2**2 + 592*K2*K3*K5 + 88*K2*K4*K6 - 1380*K3**2 - 358*K4**2 - 80*K5**2 - 18*K6**2 + 3084 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}]] |
| If K is slice | False |