| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,2,3,0,1,2,1,1,0,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1759'] |
| 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+34t^5+89t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1759'] |
| 2-strand cable arrow polynomial of the knot is: 768*K1**4*K2**3 - 1536*K1**4*K2**2 + 1504*K1**4*K2 - 1840*K1**4 - 384*K1**3*K2**2*K3 + 704*K1**3*K2*K3 + 32*K1**3*K3*K4 - 800*K1**3*K3 + 384*K1**2*K2**5 - 1984*K1**2*K2**4 + 3904*K1**2*K2**3 - 7328*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 6728*K1**2*K2 - 368*K1**2*K3**2 - 96*K1**2*K3*K5 - 32*K1**2*K4**2 - 4132*K1**2 - 384*K1*K2**4*K3 + 1824*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 - 64*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5464*K1*K2*K3 + 872*K1*K3*K4 + 200*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1832*K2**4 - 32*K2**3*K6 - 592*K2**2*K3**2 - 112*K2**2*K4**2 + 992*K2**2*K4 - 1814*K2**2 + 352*K2*K3*K5 + 24*K2*K4*K6 - 1460*K3**2 - 474*K4**2 - 136*K5**2 - 2*K6**2 + 3184 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1759'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16959', 'vk6.16963', 'vk6.17202', 'vk6.17206', 'vk6.20845', 'vk6.20849', 'vk6.22247', 'vk6.22250', 'vk6.23361', 'vk6.23657', 'vk6.23662', 'vk6.28309', 'vk6.35413', 'vk6.35835', 'vk6.35841', 'vk6.39915', 'vk6.39928', 'vk6.42013', 'vk6.43162', 'vk6.43164', 'vk6.46469', 'vk6.46478', 'vk6.55113', 'vk6.55123', 'vk6.55374', 'vk6.57656', 'vk6.57666', 'vk6.58841', 'vk6.59815', 'vk6.59835', 'vk6.68395', 'vk6.69715'] |
| 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 | O1O2O3U4U5O4O6U2U3O5U1U6 |
| R3 orbit | {'O1O2O3U4U5O4O6U2U3O5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U3O5U1U2O4O6U5U6 |
| Gauss code of K* | O1O2U3O4O5U4U1U2O6O3U6U5 |
| Gauss code of -K* | O1O2U3O4O5U1U6O3O6U4U5U2 |
| 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 -1 1 -1 0 2],[ 1 0 0 2 0 0 2],[ 1 0 0 1 1 1 1],[-1 -2 -1 0 -1 0 0],[ 1 0 -1 1 0 0 3],[ 0 0 -1 0 0 0 2],[-2 -2 -1 0 -3 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -2 -1 -2 -3],[-1 0 0 0 -1 -2 -1],[ 0 2 0 0 -1 0 0],[ 1 1 1 1 0 0 1],[ 1 2 2 0 0 0 0],[ 1 3 1 0 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,2,1,2,3,0,1,2,1,1,0,0,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,2,3,0,1,2,1,1,0,0,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,0,1,0,1,1,0,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,0,0,1,2,1,1,0,1,1,1,0,0,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,1,3,0,1,1,1,0,2,2,0,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w |
| Inner characteristic polynomial | t^6+26t^4+62t^2 |
| Outer characteristic polynomial | t^7+34t^5+89t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | 768*K1**4*K2**3 - 1536*K1**4*K2**2 + 1504*K1**4*K2 - 1840*K1**4 - 384*K1**3*K2**2*K3 + 704*K1**3*K2*K3 + 32*K1**3*K3*K4 - 800*K1**3*K3 + 384*K1**2*K2**5 - 1984*K1**2*K2**4 + 3904*K1**2*K2**3 - 7328*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 6728*K1**2*K2 - 368*K1**2*K3**2 - 96*K1**2*K3*K5 - 32*K1**2*K4**2 - 4132*K1**2 - 384*K1*K2**4*K3 + 1824*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 - 64*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5464*K1*K2*K3 + 872*K1*K3*K4 + 200*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1832*K2**4 - 32*K2**3*K6 - 592*K2**2*K3**2 - 112*K2**2*K4**2 + 992*K2**2*K4 - 1814*K2**2 + 352*K2*K3*K5 + 24*K2*K4*K6 - 1460*K3**2 - 474*K4**2 - 136*K5**2 - 2*K6**2 + 3184 |
| 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}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {4, 5}, {1, 3}, {2}]] |
| If K is slice | False |