| Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,0,2,0,1,1,0,1,1,0,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1904', '7.43701'] |
| Arrow polynomial of the knot is: -12*K1**2 + 6*K2 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.612', '6.1246', '6.1402', '6.1410', '6.1411', '6.1468', '6.1617', '6.1666', '6.1667', '6.1815', '6.1904', '6.1994', '6.1995', '6.1996', '6.2001', '6.2014', '6.2015', '6.2016', '6.2017', '6.2020', '6.2022'] |
| Outer characteristic polynomial of the knot is: t^7+14t^5+45t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1904', '6.2080'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 576*K1**4*K2 - 4704*K1**4 + 448*K1**3*K2*K3 - 864*K1**3*K3 - 5056*K1**2*K2**2 - 352*K1**2*K2*K4 + 11528*K1**2*K2 - 480*K1**2*K3**2 - 6260*K1**2 - 288*K1*K2**2*K3 + 6968*K1*K2*K3 + 696*K1*K3*K4 - 304*K2**4 + 520*K2**2*K4 - 5128*K2**2 - 2084*K3**2 - 304*K4**2 + 5214 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1904'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20010', 'vk6.20081', 'vk6.21282', 'vk6.21361', 'vk6.27061', 'vk6.27146', 'vk6.28766', 'vk6.28833', 'vk6.38450', 'vk6.38539', 'vk6.40639', 'vk6.40734', 'vk6.45334', 'vk6.45439', 'vk6.47103', 'vk6.47179', 'vk6.56809', 'vk6.56902', 'vk6.57943', 'vk6.58038', 'vk6.61327', 'vk6.61432', 'vk6.62503', 'vk6.62587', 'vk6.66521', 'vk6.66602', 'vk6.67310', 'vk6.67391', 'vk6.69167', 'vk6.69254', 'vk6.69918', 'vk6.69993'] |
| 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 | O1O2O3U4U3O5U2O6U1O4U6U5 |
| R3 orbit | {'O1O2O3U4U3O5U2O6U1O4U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O6U3O5U2O4U1U6 |
| Gauss code of K* | O1O2U3O4U5O6U1O5O3U6U4U2 |
| Gauss code of -K* | O1O2U3O4U2O5U1O6O3U6U5U4 |
| 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 0],[ 1 0 0 0 -1 2 0],[ 0 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -1],[ 1 1 1 1 0 0 0],[-1 -2 0 1 0 0 0],[ 0 0 1 1 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 0 0 -2],[-1 -1 0 0 -1 -1 0],[ 0 0 0 0 -1 -1 0],[ 0 0 1 1 0 0 0],[ 1 0 1 1 0 0 1],[ 1 2 0 0 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,1,1,-1,0,0,0,2,0,1,1,0,1,1,0,0,0,-1] |
| Phi over symmetry | [-1,-1,0,0,1,1,-1,0,0,0,2,0,1,1,0,1,1,0,0,0,-1] |
| Phi of -K | [-1,-1,0,0,1,1,-1,0,1,1,2,1,1,2,0,1,1,1,0,1,1] |
| Phi of K* | [-1,-1,0,0,1,1,-1,0,1,1,2,1,1,2,0,1,1,1,0,1,1] |
| Phi of -K* | [-1,-1,0,0,1,1,-1,0,0,0,2,0,1,1,0,1,1,0,0,0,-1] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | z^2+21z+39 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+21w^2z+39w |
| Inner characteristic polynomial | t^6+10t^4+25t^2+1 |
| Outer characteristic polynomial | t^7+14t^5+45t^3+7t |
| Flat arrow polynomial | -12*K1**2 + 6*K2 + 7 |
| 2-strand cable arrow polynomial | -64*K1**6 - 64*K1**4*K2**2 + 576*K1**4*K2 - 4704*K1**4 + 448*K1**3*K2*K3 - 864*K1**3*K3 - 5056*K1**2*K2**2 - 352*K1**2*K2*K4 + 11528*K1**2*K2 - 480*K1**2*K3**2 - 6260*K1**2 - 288*K1*K2**2*K3 + 6968*K1*K2*K3 + 696*K1*K3*K4 - 304*K2**4 + 520*K2**2*K4 - 5128*K2**2 - 2084*K3**2 - 304*K4**2 + 5214 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}]] |
| If K is slice | True |