| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,0,1,3,1,1,2,3,2,2,1,1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1001'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935'] |
| Outer characteristic polynomial of the knot is: t^7+45t^5+113t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1001'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 - 512*K1**2*K2**2 + 408*K1**2*K2 - 16*K1**2*K3**2 - 900*K1**2 + 224*K1*K2**3*K3 + 1712*K1*K2*K3 + 128*K1*K3*K4 + 32*K1*K4*K5 - 232*K2**4 - 592*K2**2*K3**2 - 16*K2**2*K4**2 + 80*K2**2*K4 - 644*K2**2 + 360*K2*K3*K5 + 32*K2*K4*K6 - 752*K3**2 - 94*K4**2 - 76*K5**2 - 12*K6**2 + 884 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1001'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70495', 'vk6.70507', 'vk6.70552', 'vk6.70572', 'vk6.70699', 'vk6.70723', 'vk6.70806', 'vk6.70824', 'vk6.70976', 'vk6.70994', 'vk6.71056', 'vk6.71080', 'vk6.71197', 'vk6.71213', 'vk6.71274', 'vk6.71286', 'vk6.71743', 'vk6.72164', 'vk6.73815', 'vk6.73816', 'vk6.73958', 'vk6.73967', 'vk6.73970', 'vk6.75771', 'vk6.75792', 'vk6.75799', 'vk6.77546', 'vk6.78784', 'vk6.78791', 'vk6.79070', 'vk6.80370', 'vk6.80381', 'vk6.81257', 'vk6.81784', 'vk6.87005', 'vk6.87802', 'vk6.87937', 'vk6.89140', 'vk6.89149', 'vk6.89340'] |
| 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 | O1O2O3O4U2U1O5U4O6U5U3U6 |
| R3 orbit | {'O1O2O3O4U2U1O5U4O6U5U3U6', 'O1O2O3O4U2U1U3O5O6U4U5U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U2U6O5U1O6U4U3 |
| Gauss code of K* | O1O2O3U4U5U2U6O5O4U1O6U3 |
| Gauss code of -K* | O1O2O3U1O4U3O5O6U4U2U6U5 |
| 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 -2 1 1 0 2],[ 2 0 0 3 2 1 1],[ 2 0 0 2 1 1 1],[-1 -3 -2 0 -1 1 2],[-1 -2 -1 1 0 1 1],[ 0 -1 -1 -1 -1 0 1],[-2 -1 -1 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 -1 -2 -1 -1 -1],[-1 1 0 1 1 -1 -2],[-1 2 -1 0 1 -2 -3],[ 0 1 -1 -1 0 -1 -1],[ 2 1 1 2 1 0 0],[ 2 1 2 3 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,1,2,1,1,1,-1,-1,1,2,-1,2,3,1,1,0] |
| Phi over symmetry | [-2,-2,0,1,1,2,0,1,0,1,3,1,1,2,3,2,2,1,1,-1,0] |
| Phi of -K | [-2,-2,0,1,1,2,0,1,0,1,3,1,1,2,3,2,2,1,1,-1,0] |
| Phi of K* | [-2,-1,-1,0,2,2,-1,0,1,3,3,-1,2,0,1,2,1,2,1,1,0] |
| Phi of -K* | [-2,-2,0,1,1,2,0,1,1,2,1,1,2,3,1,-1,-1,1,1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 3z+7 |
| Enhanced Jones-Krushkal polynomial | 4w^4z-10w^3z+4w^3+9w^2z+3w |
| Inner characteristic polynomial | t^6+31t^4+20t^2 |
| Outer characteristic polynomial | t^7+45t^5+113t^3 |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 - 512*K1**2*K2**2 + 408*K1**2*K2 - 16*K1**2*K3**2 - 900*K1**2 + 224*K1*K2**3*K3 + 1712*K1*K2*K3 + 128*K1*K3*K4 + 32*K1*K4*K5 - 232*K2**4 - 592*K2**2*K3**2 - 16*K2**2*K4**2 + 80*K2**2*K4 - 644*K2**2 + 360*K2*K3*K5 + 32*K2*K4*K6 - 752*K3**2 - 94*K4**2 - 76*K5**2 - 12*K6**2 + 884 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {4, 5}, {3}, {1, 2}]] |
| If K is slice | False |