| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,0,1,2,1,1,0,1,1,0,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1235', '7.41173'] |
| 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+18t^5+40t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1235', '7.41173'] |
| 2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 3136*K1**4 + 768*K1**3*K2*K3 - 320*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4144*K1**2*K2**2 - 384*K1**2*K2*K4 + 3512*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K4**2 + 208*K1**2 + 320*K1*K2**3*K3 - 864*K1*K2**2*K3 - 192*K1*K2**2*K5 + 2656*K1*K2*K3 + 520*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 632*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 632*K2**2*K4 - 694*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 412*K3**2 - 146*K4**2 - 36*K5**2 - 2*K6**2 + 840 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1235'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.410', 'vk6.460', 'vk6.464', 'vk6.859', 'vk6.912', 'vk6.917', 'vk6.1614', 'vk6.2095', 'vk6.2464', 'vk6.2499', 'vk6.2502', 'vk6.2711', 'vk6.2755', 'vk6.2759', 'vk6.3018', 'vk6.3151', 'vk6.3222', 'vk6.3243', 'vk6.3323', 'vk6.3338', 'vk6.3355', 'vk6.3444', 'vk6.15226', 'vk6.15229', 'vk6.15232', 'vk6.15239', 'vk6.15256', 'vk6.15259', 'vk6.26357', 'vk6.26360', 'vk6.26800', 'vk6.26805', 'vk6.33875', 'vk6.33890', 'vk6.33898', 'vk6.37938', 'vk6.37946', 'vk6.45097', 'vk6.45102', 'vk6.54444'] |
| 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 | O1O2O3O4U3U4U2O5O6U5U1U6 |
| R3 orbit | {'O1O2O3U2O4U3U4O5O6U5U1U6', 'O1O2O3O4U3U4U2O5O6U5U1U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U4U6O5O6U3U1U2 |
| Gauss code of K* | O1O2O3U4U5O4O6O5U6U3U1U2 |
| Gauss code of -K* | O1O2O3U1U3O4O5O6U5U6U4U2 |
| 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 -1 1 -1 2],[ 1 0 0 -1 1 0 2],[ 0 0 0 -1 1 0 0],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 1 0 0 0 0 0 1],[-2 -2 0 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 0 0 -1 -2],[-1 0 0 -1 -1 0 -1],[ 0 0 1 0 -1 0 0],[ 1 0 1 1 0 0 1],[ 1 1 0 0 0 0 0],[ 1 2 1 0 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,0,0,1,2,1,1,0,1,1,0,0,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,0,1,2,1,1,0,1,1,0,0,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,3,0,1,1,1,1,2,2,0,2,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,2,1,2,3,0,1,2,1,1,1,0,0,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,0,0,0,1,1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+19w^2z+15w |
| Inner characteristic polynomial | t^6+10t^4+17t^2 |
| Outer characteristic polynomial | t^7+18t^5+40t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 1536*K1**4*K2 - 3136*K1**4 + 768*K1**3*K2*K3 - 320*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4144*K1**2*K2**2 - 384*K1**2*K2*K4 + 3512*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K4**2 + 208*K1**2 + 320*K1*K2**3*K3 - 864*K1*K2**2*K3 - 192*K1*K2**2*K5 + 2656*K1*K2*K3 + 520*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 632*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 632*K2**2*K4 - 694*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 412*K3**2 - 146*K4**2 - 36*K5**2 - 2*K6**2 + 840 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}]] |
| If K is slice | False |