| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,0,1,0,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1800'] |
| 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+20t^5+37t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1800'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 128*K1**4*K2**2 + 2016*K1**4*K2 - 5008*K1**4 + 256*K1**3*K2*K3 - 608*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5264*K1**2*K2**2 - 416*K1**2*K2*K4 + 9936*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 5416*K1**2 + 448*K1*K2**3*K3 - 960*K1*K2**2*K3 - 64*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 6432*K1*K2*K3 + 1256*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 712*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 1256*K2**2*K4 - 4934*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 2248*K3**2 - 754*K4**2 - 72*K5**2 - 2*K6**2 + 5144 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1800'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13875', 'vk6.13972', 'vk6.14143', 'vk6.14366', 'vk6.14950', 'vk6.15073', 'vk6.15595', 'vk6.16065', 'vk6.16286', 'vk6.16309', 'vk6.17428', 'vk6.22597', 'vk6.22628', 'vk6.23936', 'vk6.33686', 'vk6.33766', 'vk6.34151', 'vk6.34252', 'vk6.34587', 'vk6.36211', 'vk6.36240', 'vk6.42281', 'vk6.53853', 'vk6.53899', 'vk6.54100', 'vk6.54394', 'vk6.54571', 'vk6.55583', 'vk6.59022', 'vk6.59039', 'vk6.60070', 'vk6.64564'] |
| 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 | O1O2O3U4U3O5O6U1U5O4U6U2 |
| R3 orbit | {'O1O2O3U4U3O5O6U1U5O4U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U4O5U6U3O4O6U1U5 |
| Gauss code of K* | O1O2U3O4O5U1U5U6O3O6U2U4 |
| Gauss code of -K* | O1O2U3O4O5U2U4O6O3U6U1U5 |
| 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 1 1 -1 0 1],[ 2 0 2 0 0 1 1],[-1 -2 0 1 -1 -1 0],[-1 0 -1 0 -1 0 -1],[ 1 0 1 1 0 0 0],[ 0 -1 1 0 0 0 1],[-1 -1 0 1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 0 -1],[-1 -1 0 -1 0 -1 0],[-1 0 1 0 -1 -1 -2],[ 0 1 0 1 0 0 -1],[ 1 0 1 1 0 0 0],[ 2 1 0 2 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,0,1,1,0,1,0,1,1,2,0,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,0,1,2,0,1,0,1,0,1,1,-1,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,1,2,3,1,1,2,1,0,0,1,0,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,1,1,3,0,0,1,1,0,2,2,1,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,0,1,2,0,1,0,1,0,1,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+12t^4+16t^2 |
| Outer characteristic polynomial | t^7+20t^5+37t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 - 128*K1**4*K2**2 + 2016*K1**4*K2 - 5008*K1**4 + 256*K1**3*K2*K3 - 608*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5264*K1**2*K2**2 - 416*K1**2*K2*K4 + 9936*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 5416*K1**2 + 448*K1*K2**3*K3 - 960*K1*K2**2*K3 - 64*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 6432*K1*K2*K3 + 1256*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 712*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 1256*K2**2*K4 - 4934*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 2248*K3**2 - 754*K4**2 - 72*K5**2 - 2*K6**2 + 5144 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |