| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,1,1,0,0,-1,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.806', '7.38450'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+32t^5+36t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.806', '6.1201'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 320*K1**4*K2**2 + 960*K1**4*K2 - 2320*K1**4 + 384*K1**3*K2*K3 - 480*K1**3*K3 - 192*K1**2*K2**4 + 736*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2768*K1**2*K2**2 - 544*K1**2*K2*K4 + 3328*K1**2*K2 - 176*K1**2*K3**2 - 48*K1**2*K4**2 - 364*K1**2 + 224*K1*K2**3*K3 - 224*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2088*K1*K2*K3 + 248*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 376*K2**4 - 32*K2**3*K6 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 368*K2**2*K4 - 734*K2**2 + 72*K2*K3*K5 + 16*K2*K4*K6 - 284*K3**2 - 90*K4**2 - 16*K5**2 - 2*K6**2 + 832 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.806'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.320', 'vk6.357', 'vk6.419', 'vk6.715', 'vk6.760', 'vk6.827', 'vk6.866', 'vk6.1126', 'vk6.1501', 'vk6.1567', 'vk6.1666', 'vk6.1950', 'vk6.1987', 'vk6.2040', 'vk6.2175', 'vk6.2282', 'vk6.2647', 'vk6.2730', 'vk6.2788', 'vk6.3110', 'vk6.5245', 'vk6.6500', 'vk6.8874', 'vk6.9789', 'vk6.18319', 'vk6.18656', 'vk6.19405', 'vk6.19698', 'vk6.25209', 'vk6.25863', 'vk6.26187', 'vk6.28496', 'vk6.36932', 'vk6.37396', 'vk6.37970', 'vk6.39872', 'vk6.40276', 'vk6.44852', 'vk6.46940', 'vk6.49133'] |
| 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 | O1O2O3O4U5O6U4U6U3O5U1U2 |
| R3 orbit | {'O1O2O3O4U5U3O6U4U6O5U1U2', 'O1O2O3O4U5O6U4U6U3O5U1U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U3U4O5U2U6U1O6U5 |
| Gauss code of K* | O1O2O3U4O5O6U5U6U3U1O4U2 |
| Gauss code of -K* | O1O2O3U2O4U3U1U5U6O5O6U4 |
| 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 0 -2 1],[ 1 0 1 1 0 -1 1],[-1 -1 0 1 0 -3 1],[-1 -1 -1 0 -1 -2 1],[ 0 0 0 1 0 -1 1],[ 2 1 3 2 1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -3],[-1 -1 0 1 -1 -1 -2],[-1 -1 -1 0 -1 -1 -1],[ 0 0 1 1 0 0 -1],[ 1 1 1 1 0 0 -1],[ 2 3 2 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,0,1,3,-1,1,1,2,1,1,1,0,1,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,1,1,0,0,-1,-1,-1] |
| Phi of -K | [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,1,1,0,0,-1,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,0,1,2,-1,0,1,1,1,1,0,1,1,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,1,1,2,3,0,1,1,1,1,1,0,-1,-1,-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+24t^4+17t^2 |
| Outer characteristic polynomial | t^7+32t^5+36t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -256*K1**6 - 320*K1**4*K2**2 + 960*K1**4*K2 - 2320*K1**4 + 384*K1**3*K2*K3 - 480*K1**3*K3 - 192*K1**2*K2**4 + 736*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2768*K1**2*K2**2 - 544*K1**2*K2*K4 + 3328*K1**2*K2 - 176*K1**2*K3**2 - 48*K1**2*K4**2 - 364*K1**2 + 224*K1*K2**3*K3 - 224*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2088*K1*K2*K3 + 248*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 376*K2**4 - 32*K2**3*K6 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 368*K2**2*K4 - 734*K2**2 + 72*K2*K3*K5 + 16*K2*K4*K6 - 284*K3**2 - 90*K4**2 - 16*K5**2 - 2*K6**2 + 832 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |