| Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,1,2,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1617'] |
| 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+23t^5+31t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1617'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**6 - 256*K1**4*K2**2 + 1760*K1**4*K2 - 4096*K1**4 + 576*K1**3*K2*K3 - 832*K1**3*K3 - 4256*K1**2*K2**2 - 160*K1**2*K2*K4 + 7880*K1**2*K2 - 480*K1**2*K3**2 - 3272*K1**2 - 128*K1*K2**2*K3 + 4312*K1*K2*K3 + 432*K1*K3*K4 - 112*K2**4 + 96*K2**2*K4 - 2936*K2**2 - 1096*K3**2 - 100*K4**2 + 3050 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1617'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10960', 'vk6.10961', 'vk6.10992', 'vk6.10993', 'vk6.12126', 'vk6.12127', 'vk6.12158', 'vk6.12159', 'vk6.13783', 'vk6.13806', 'vk6.14218', 'vk6.14226', 'vk6.14667', 'vk6.14675', 'vk6.14858', 'vk6.14877', 'vk6.15821', 'vk6.15829', 'vk6.31825', 'vk6.31826', 'vk6.33615', 'vk6.33644', 'vk6.33646', 'vk6.33677', 'vk6.51805', 'vk6.51806', 'vk6.52667', 'vk6.52668', 'vk6.53809', 'vk6.53816', 'vk6.54228', 'vk6.54236'] |
| 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 | O1O2O3U2O4U5U4O6U3O5U1U6 |
| R3 orbit | {'O1O2O3U2O4U5U4O6U3O5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U3O5U1O4U6U5O6U2 |
| Gauss code of K* | O1O2U3O4U1O5O3U5U6U4O6U2 |
| Gauss code of -K* | O1O2U3O4U1O5O3U5O6U4U6U2 |
| 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 1 1 -1 1],[ 1 0 -1 2 1 -1 1],[ 1 1 0 1 0 0 1],[-1 -2 -1 0 0 -2 0],[-1 -1 0 0 0 -1 -1],[ 1 1 0 2 1 0 1],[-1 -1 -1 0 1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 -1 -1 -1],[-1 0 1 0 -1 -1 -1],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 -1 -2 -2],[ 1 1 0 1 0 1 0],[ 1 1 1 2 -1 0 -1],[ 1 1 1 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,1,1,1,-1,0,1,1,1,0,0,1,1,1,2,2,-1,0,1] |
| Phi over symmetry | [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,1,2,-1,0,0] |
| Phi of -K | [-1,-1,-1,1,1,1,-1,0,0,1,1,1,0,1,1,1,1,2,0,0,-1] |
| Phi of K* | [-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,1,0,0,1,-1,-1,0] |
| Phi of -K* | [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,1,2,-1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 17z+35 |
| Enhanced Jones-Krushkal polynomial | 17w^2z+35w |
| Inner characteristic polynomial | t^6+17t^4+17t^2+1 |
| Outer characteristic polynomial | t^7+23t^5+31t^3+5t |
| Flat arrow polynomial | -12*K1**2 + 6*K2 + 7 |
| 2-strand cable arrow polynomial | -384*K1**6 - 256*K1**4*K2**2 + 1760*K1**4*K2 - 4096*K1**4 + 576*K1**3*K2*K3 - 832*K1**3*K3 - 4256*K1**2*K2**2 - 160*K1**2*K2*K4 + 7880*K1**2*K2 - 480*K1**2*K3**2 - 3272*K1**2 - 128*K1*K2**2*K3 + 4312*K1*K2*K3 + 432*K1*K3*K4 - 112*K2**4 + 96*K2**2*K4 - 2936*K2**2 - 1096*K3**2 - 100*K4**2 + 3050 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}]] |
| If K is slice | True |