| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,2,-1,1,1,0,1,1,1,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1493'] |
| 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+23t^5+46t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1493'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 256*K1**2*K2**3 - 1008*K1**2*K2**2 + 1136*K1**2*K2 - 48*K1**2*K3**2 - 672*K1**2 + 64*K1*K2**3*K3 + 736*K1*K2*K3 + 136*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 280*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 192*K2**2*K4 - 454*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 244*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 660 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1493'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11288', 'vk6.11366', 'vk6.12551', 'vk6.12662', 'vk6.13379', 'vk6.13456', 'vk6.13645', 'vk6.13759', 'vk6.14154', 'vk6.14385', 'vk6.15616', 'vk6.16080', 'vk6.16472', 'vk6.16487', 'vk6.17634', 'vk6.18691', 'vk6.22883', 'vk6.22914', 'vk6.24793', 'vk6.25252', 'vk6.30973', 'vk6.31099', 'vk6.32152', 'vk6.32271', 'vk6.33134', 'vk6.33181', 'vk6.33243', 'vk6.33292', 'vk6.34864', 'vk6.34895', 'vk6.36983', 'vk6.37437', 'vk6.39605', 'vk6.41846', 'vk6.42446', 'vk6.44165', 'vk6.46221', 'vk6.47828', 'vk6.52060', 'vk6.52499', 'vk6.52897', 'vk6.53373', 'vk6.53555', 'vk6.53596', 'vk6.53627', 'vk6.53687', 'vk6.56351', 'vk6.60985'] |
| 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 | O1O2O3U1O4U3U5O6O5U4U2U6 |
| R3 orbit | {'O1O2O3U1O4U3U5O6O5U4U2U6', 'O1O2O3U1U2U4O5O4O6U3U6U5', 'O1O2O3U1U2O4U5O6O5U3U4U6'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3U4U2U5O6O4U6U1O5U3 |
| Gauss code of K* | O1O2O3U4U2U5O4U1O5O6U3U6 |
| Gauss code of -K* | O1O2O3U4U1O4O5U3O6U5U2U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 0 0 0 1 1],[ 2 0 2 1 1 2 1],[ 0 -2 0 -1 1 0 1],[ 0 -1 1 0 1 0 1],[ 0 -1 -1 -1 0 0 0],[-1 -2 0 0 0 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 1 0 0 0 -2],[-1 -1 0 0 -1 -1 -1],[ 0 0 0 0 -1 -1 -1],[ 0 0 1 1 0 1 -1],[ 0 0 1 1 -1 0 -2],[ 2 2 1 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,-1,0,0,0,2,0,1,1,1,1,1,1,-1,1,2] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,1,1,2,-1,1,1,0,1,1,1,1,0,-1] |
| Phi of -K | [-2,0,0,0,1,1,0,1,1,1,2,-1,1,1,0,1,1,1,1,0,-1] |
| Phi of K* | [-1,-1,0,0,0,2,-1,0,0,1,2,1,1,1,1,-1,1,0,1,1,1] |
| Phi of -K* | [-2,0,0,0,1,1,1,1,2,1,2,-1,-1,0,0,1,1,0,1,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+9w^2z+15w |
| Inner characteristic polynomial | t^6+17t^4+17t^2 |
| Outer characteristic polynomial | t^7+23t^5+46t^3 |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 256*K1**2*K2**3 - 1008*K1**2*K2**2 + 1136*K1**2*K2 - 48*K1**2*K3**2 - 672*K1**2 + 64*K1*K2**3*K3 + 736*K1*K2*K3 + 136*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 280*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 192*K2**2*K4 - 454*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 244*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 660 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |