| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,1,1,0,0,1,0,1,0,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1791'] |
| 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+16t^5+32t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1791'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 800*K1**4 + 384*K1**3*K2*K3 - 256*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 - 4320*K1**2*K2**2 - 352*K1**2*K2*K4 + 5504*K1**2*K2 - 160*K1**2*K3**2 - 3932*K1**2 + 352*K1*K2**3*K3 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4568*K1*K2*K3 + 416*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 536*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 744*K2**2*K4 - 2734*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1212*K3**2 - 282*K4**2 - 24*K5**2 - 2*K6**2 + 2808 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1791'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11433', 'vk6.11729', 'vk6.12744', 'vk6.13088', 'vk6.20331', 'vk6.21672', 'vk6.27631', 'vk6.29175', 'vk6.31187', 'vk6.31528', 'vk6.32351', 'vk6.32769', 'vk6.39063', 'vk6.41321', 'vk6.45815', 'vk6.47486', 'vk6.52186', 'vk6.52444', 'vk6.53014', 'vk6.53330', 'vk6.57202', 'vk6.58417', 'vk6.61812', 'vk6.62937', 'vk6.63758', 'vk6.63869', 'vk6.64183', 'vk6.64370', 'vk6.66815', 'vk6.67683', 'vk6.69451', 'vk6.70173'] |
| 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 | O1O2O3U4U1O5O6U5U2O4U6U3 |
| R3 orbit | {'O1O2O3U4U1O5O6U5U2O4U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U2U6O4O6U3U5 |
| Gauss code of K* | O1O2U3O4O5U6U2U5O3O6U1U4 |
| Gauss code of -K* | O1O2U3O4O5U2U5O6O3U1U4U6 |
| 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 2 -1 -1 1],[ 1 0 1 1 0 0 0],[ 0 -1 0 1 0 0 1],[-2 -1 -1 0 -1 -1 0],[ 1 0 0 1 0 -1 0],[ 1 0 0 1 1 0 1],[-1 0 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -1 -1],[-1 0 0 -1 0 0 -1],[ 0 1 1 0 0 -1 0],[ 1 1 0 0 0 0 -1],[ 1 1 0 1 0 0 0],[ 1 1 1 0 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,1,1,1,1,1,0,0,1,0,1,0,0,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,1,1,1,0,0,1,0,1,0,0,1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,2,2,0,2,2,0,1,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,1,2,2,2,0,1,2,2,1,0,1,0,1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,0,1,0,0,1,1,1,0,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+8t^4+11t^2 |
| Outer characteristic polynomial | t^7+16t^5+32t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 480*K1**4*K2 - 800*K1**4 + 384*K1**3*K2*K3 - 256*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 - 4320*K1**2*K2**2 - 352*K1**2*K2*K4 + 5504*K1**2*K2 - 160*K1**2*K3**2 - 3932*K1**2 + 352*K1*K2**3*K3 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4568*K1*K2*K3 + 416*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 536*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 744*K2**2*K4 - 2734*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1212*K3**2 - 282*K4**2 - 24*K5**2 - 2*K6**2 + 2808 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | False |