| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,0,0,0,0,1,1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1550'] |
| 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+40t^5+176t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1550'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 + 256*K1**4*K2**3 - 768*K1**4*K2**2 + 2176*K1**4*K2 - 3248*K1**4 - 384*K1**3*K2**2*K3 + 1152*K1**3*K2*K3 - 512*K1**3*K3 - 448*K1**2*K2**4 + 1088*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5760*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 7952*K1**2*K2 - 624*K1**2*K3**2 - 4252*K1**2 + 768*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5504*K1*K2*K3 + 512*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 936*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 872*K2**2*K4 - 3406*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 1452*K3**2 - 162*K4**2 - 8*K5**2 - 2*K6**2 + 3632 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1550'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11034', 'vk6.11114', 'vk6.12202', 'vk6.12311', 'vk6.16419', 'vk6.19236', 'vk6.19333', 'vk6.19529', 'vk6.19626', 'vk6.22722', 'vk6.22823', 'vk6.26048', 'vk6.26101', 'vk6.26428', 'vk6.26523', 'vk6.30611', 'vk6.30708', 'vk6.31919', 'vk6.34766', 'vk6.38117', 'vk6.38135', 'vk6.42383', 'vk6.44643', 'vk6.44763', 'vk6.51831', 'vk6.52703', 'vk6.52799', 'vk6.56581', 'vk6.56639', 'vk6.64712', 'vk6.66281', 'vk6.66305'] |
| 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 | O1O2O3U4O5U2U6O4O6U3U1U5 |
| R3 orbit | {'O1O2O3U4O5U2U6O4O6U3U1U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U3U1O5O6U5U2O4U6 |
| Gauss code of K* | O1O2O3U2U4U1O5U3O4O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O6U1O5U3U6U2 |
| 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 0 -1 2 1],[ 1 0 0 1 -1 2 2],[ 1 0 0 0 1 1 2],[ 0 -1 0 0 -1 0 1],[ 1 1 -1 1 0 3 1],[-2 -2 -1 0 -3 0 -2],[-1 -2 -2 -1 -1 2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -2 0 -1 -2 -3],[-1 2 0 -1 -2 -2 -1],[ 0 0 1 0 0 -1 -1],[ 1 1 2 0 0 0 1],[ 1 2 2 1 0 0 -1],[ 1 3 1 1 -1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,2,0,1,2,3,1,2,2,1,0,1,1,0,-1,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,0,0,0,0,1,1,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,1,0,2,-1,0,1,0,0,0,1,0,2,-1] |
| Phi of K* | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,0,0,0,0,1,1,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,2,2,-1,1,1,3,0,2,1,1,0,2] |
| 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+32t^4+133t^2+4 |
| Outer characteristic polynomial | t^7+40t^5+176t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -256*K1**6 + 256*K1**4*K2**3 - 768*K1**4*K2**2 + 2176*K1**4*K2 - 3248*K1**4 - 384*K1**3*K2**2*K3 + 1152*K1**3*K2*K3 - 512*K1**3*K3 - 448*K1**2*K2**4 + 1088*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5760*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 7952*K1**2*K2 - 624*K1**2*K3**2 - 4252*K1**2 + 768*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5504*K1*K2*K3 + 512*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 936*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 872*K2**2*K4 - 3406*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 1452*K3**2 - 162*K4**2 - 8*K5**2 - 2*K6**2 + 3632 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |