| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,2,1,1,1,1,0,1,-1,0,1,0,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1491'] |
| Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717'] |
| Outer characteristic polynomial of the knot is: t^7+25t^5+58t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1491'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1376*K1**4*K2 - 4336*K1**4 + 480*K1**3*K2*K3 - 1184*K1**3*K3 + 128*K1**2*K2**3 - 4016*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 10176*K1**2*K2 - 1552*K1**2*K3**2 - 64*K1**2*K3*K5 - 6772*K1**2 - 864*K1*K2**2*K3 - 320*K1*K2*K3*K4 + 7800*K1*K2*K3 + 2080*K1*K3*K4 + 272*K1*K4*K5 - 128*K2**4 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 744*K2**2*K4 - 5636*K2**2 + 536*K2*K3*K5 + 32*K2*K4*K6 - 32*K3**4 + 80*K3**2*K6 - 3020*K3**2 - 836*K4**2 - 256*K5**2 - 44*K6**2 + 5882 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1491'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3559', 'vk6.3580', 'vk6.3584', 'vk6.3805', 'vk6.3809', 'vk6.3838', 'vk6.3842', 'vk6.6967', 'vk6.6979', 'vk6.7000', 'vk6.7012', 'vk6.7189', 'vk6.7201', 'vk6.7222', 'vk6.15344', 'vk6.15347', 'vk6.15471', 'vk6.15472', 'vk6.33983', 'vk6.34032', 'vk6.34035', 'vk6.34441', 'vk6.48208', 'vk6.48220', 'vk6.48370', 'vk6.49949', 'vk6.49970', 'vk6.49974', 'vk6.54008', 'vk6.54009', 'vk6.54056', 'vk6.54503'] |
| 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 | O1O2O3U1O4U3U4O5O6U5U2U6 |
| R3 orbit | {'O1O2O3U1O4U3U4O5O6U5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2U5O4O5U6U1O6U3 |
| Gauss code of K* | O1O2O3U4U2U5O4U6O5O6U1U3 |
| Gauss code of -K* | O1O2O3U1U3O4O5U4O6U5U2U6 |
| 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 -2 0 0 1 -1 2],[ 2 0 2 1 1 0 1],[ 0 -2 0 -1 1 0 2],[ 0 -1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -2 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -2 -1 -1],[-1 0 0 -1 -1 0 -1],[ 0 0 1 0 1 0 -1],[ 0 2 1 -1 0 0 -2],[ 1 1 0 0 0 0 0],[ 2 1 1 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,2,1,1,1,1,0,1,-1,0,1,0,2,0] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,2,1,1,1,1,0,1,-1,0,1,0,2,0] |
| Phi of -K | [-2,-1,0,0,1,2,1,0,1,2,3,1,1,2,2,1,0,0,0,2,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,0,2,2,3,0,0,2,2,-1,1,0,1,1,1] |
| Phi of -K* | [-2,-1,0,0,1,2,0,1,2,1,1,0,0,0,1,1,1,0,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 21z+43 |
| Enhanced Jones-Krushkal polynomial | 21w^2z+43w |
| Inner characteristic polynomial | t^6+15t^4+24t^2 |
| Outer characteristic polynomial | t^7+25t^5+58t^3+4t |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 1376*K1**4*K2 - 4336*K1**4 + 480*K1**3*K2*K3 - 1184*K1**3*K3 + 128*K1**2*K2**3 - 4016*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 10176*K1**2*K2 - 1552*K1**2*K3**2 - 64*K1**2*K3*K5 - 6772*K1**2 - 864*K1*K2**2*K3 - 320*K1*K2*K3*K4 + 7800*K1*K2*K3 + 2080*K1*K3*K4 + 272*K1*K4*K5 - 128*K2**4 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 744*K2**2*K4 - 5636*K2**2 + 536*K2*K3*K5 + 32*K2*K4*K6 - 32*K3**4 + 80*K3**2*K6 - 3020*K3**2 - 836*K4**2 - 256*K5**2 - 44*K6**2 + 5882 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}]] |
| If K is slice | False |