| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,1,1,0,2,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1702'] |
| 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+22t^5+54t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1702', '6.1805'] |
| 2-strand cable arrow polynomial of the knot is: 3008*K1**4*K2 - 6432*K1**4 + 1600*K1**3*K2*K3 - 1184*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 7008*K1**2*K2**2 - 640*K1**2*K2*K4 + 9288*K1**2*K2 - 1824*K1**2*K3**2 - 96*K1**2*K4**2 - 2596*K1**2 + 320*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 256*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6816*K1*K2*K3 + 1816*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 1040*K2**2*K4 - 3574*K2**2 + 344*K2*K3*K5 + 16*K2*K4*K6 - 1744*K3**2 - 558*K4**2 - 92*K5**2 - 2*K6**2 + 3612 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1702'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4351', 'vk6.4384', 'vk6.5673', 'vk6.5706', 'vk6.7742', 'vk6.7775', 'vk6.9224', 'vk6.9257', 'vk6.10481', 'vk6.10536', 'vk6.10633', 'vk6.10702', 'vk6.10735', 'vk6.10822', 'vk6.14615', 'vk6.15312', 'vk6.15439', 'vk6.16234', 'vk6.17981', 'vk6.24423', 'vk6.30168', 'vk6.30223', 'vk6.30320', 'vk6.30449', 'vk6.33958', 'vk6.34359', 'vk6.34415', 'vk6.43858', 'vk6.50429', 'vk6.50461', 'vk6.54217', 'vk6.63428'] |
| 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 | O1O2O3U1U3O4O5U4U2O6U5U6 |
| R3 orbit | {'O1O2O3U1U3O4O5U4U2O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U2U6O5O6U1U3 |
| Gauss code of K* | O1O2U3O4O3U5U2U6O5O6U1U4 |
| Gauss code of -K* | O1O2U1O3O4U2U4O5O6U5U3U6 |
| 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 0 1 -1 1 1],[ 2 0 2 1 0 1 0],[ 0 -2 0 0 0 2 1],[-1 -1 0 0 0 0 0],[ 1 0 0 0 0 1 1],[-1 -1 -2 0 -1 0 1],[-1 0 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -2 -1 -1],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 0 0 -1],[ 0 2 1 0 0 0 -2],[ 1 1 1 0 0 0 0],[ 2 1 0 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,2,1,1,0,1,1,0,0,0,1,0,2,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,1,1,0,2,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,0,2,2,3,1,1,2,1,-1,1,0,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,3,0,-1,1,2,1,2,2,1,0,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,1,1,0,2,0,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+14t^4+19t^2+1 |
| Outer characteristic polynomial | t^7+22t^5+54t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 3008*K1**4*K2 - 6432*K1**4 + 1600*K1**3*K2*K3 - 1184*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 7008*K1**2*K2**2 - 640*K1**2*K2*K4 + 9288*K1**2*K2 - 1824*K1**2*K3**2 - 96*K1**2*K4**2 - 2596*K1**2 + 320*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 256*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6816*K1*K2*K3 + 1816*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 1040*K2**2*K4 - 3574*K2**2 + 344*K2*K3*K5 + 16*K2*K4*K6 - 1744*K3**2 - 558*K4**2 - 92*K5**2 - 2*K6**2 + 3612 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]] |
| If K is slice | False |