| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,2,2,0,1,1,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.753'] |
| 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+41t^5+34t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.753'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 160*K1**4*K2 - 464*K1**4 + 96*K1**3*K2*K3 + 448*K1**2*K2**3 - 2736*K1**2*K2**2 - 96*K1**2*K2*K4 + 3104*K1**2*K2 - 80*K1**2*K3**2 - 1816*K1**2 + 96*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 2144*K1*K2*K3 + 216*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 536*K2**4 - 32*K2**3*K6 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 568*K2**2*K4 - 1190*K2**2 + 96*K2*K3*K5 + 16*K2*K4*K6 - 436*K3**2 - 154*K4**2 - 20*K5**2 - 2*K6**2 + 1312 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.753'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10132', 'vk6.10185', 'vk6.10330', 'vk6.10421', 'vk6.17660', 'vk6.17709', 'vk6.24227', 'vk6.24276', 'vk6.24787', 'vk6.25246', 'vk6.29915', 'vk6.29958', 'vk6.30023', 'vk6.30076', 'vk6.30971', 'vk6.31098', 'vk6.32153', 'vk6.32274', 'vk6.36587', 'vk6.36973', 'vk6.43588', 'vk6.43700', 'vk6.51708', 'vk6.51727', 'vk6.52057', 'vk6.52139', 'vk6.55698', 'vk6.55757', 'vk6.60268', 'vk6.60332', 'vk6.60652', 'vk6.60995', 'vk6.63337', 'vk6.63366', 'vk6.63390', 'vk6.63406', 'vk6.65449', 'vk6.65787', 'vk6.68548', 'vk6.68581'] |
| 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 | O1O2O3O4U2O5U4U1U5O6U3U6 |
| R3 orbit | {'O1O2O3U1O4O5U2U4U5O6U3U6', 'O1O2O3O4U2O5U4U1U5O6U3U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U2O5U6U4U1O6U3 |
| Gauss code of K* | O1O2O3U4O5O4U2U6U5U1O6U3 |
| Gauss code of -K* | O1O2O3U1O4U3U5U4U2O6O5U6 |
| 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 -2 1 0 2 1],[ 2 0 -1 3 1 2 1],[ 2 1 0 2 1 1 1],[-1 -3 -2 0 -1 1 1],[ 0 -1 -1 1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 0 -1 -1 -1 -2],[-1 0 0 -1 0 -1 -1],[-1 1 1 0 -1 -2 -3],[ 0 1 0 1 0 -1 -1],[ 2 1 1 2 1 0 1],[ 2 2 1 3 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,0,1,1,1,2,1,0,1,1,1,2,3,1,1,-1] |
| Phi over symmetry | [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,2,2,0,1,1,-1,0,1] |
| Phi of -K | [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,2,2,0,1,1,-1,0,1] |
| Phi of K* | [-2,-1,-1,0,2,2,0,1,1,2,3,1,0,0,1,1,2,2,1,1,-1] |
| Phi of -K* | [-2,-2,0,1,1,2,-1,1,1,3,2,1,1,2,1,0,1,1,-1,0,1] |
| 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+27t^4+17t^2 |
| Outer characteristic polynomial | t^7+41t^5+34t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 160*K1**4*K2 - 464*K1**4 + 96*K1**3*K2*K3 + 448*K1**2*K2**3 - 2736*K1**2*K2**2 - 96*K1**2*K2*K4 + 3104*K1**2*K2 - 80*K1**2*K3**2 - 1816*K1**2 + 96*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 2144*K1*K2*K3 + 216*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 536*K2**4 - 32*K2**3*K6 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 568*K2**2*K4 - 1190*K2**2 + 96*K2*K3*K5 + 16*K2*K4*K6 - 436*K3**2 - 154*K4**2 - 20*K5**2 - 2*K6**2 + 1312 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {3, 5}, {2, 4}, {1}]] |
| If K is slice | False |