| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,2,2,1,0,1,1,1,0,1,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.472'] |
| Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354'] |
| Outer characteristic polynomial of the knot is: t^7+54t^5+79t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.472'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 256*K1**4*K2 - 1248*K1**4 + 96*K1**3*K2*K3 - 96*K1**3*K3 - 1264*K1**2*K2**2 - 64*K1**2*K2*K4 + 2904*K1**2*K2 - 224*K1**2*K3**2 - 32*K1**2*K3*K5 - 1896*K1**2 - 224*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 1952*K1*K2*K3 + 664*K1*K3*K4 + 152*K1*K4*K5 + 24*K1*K5*K6 - 88*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 416*K2**2*K4 - 1718*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 868*K3**2 - 446*K4**2 - 148*K5**2 - 18*K6**2 + 1844 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.472'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4423', 'vk6.4520', 'vk6.4849', 'vk6.5192', 'vk6.5809', 'vk6.5938', 'vk6.6422', 'vk6.6425', 'vk6.6852', 'vk6.7983', 'vk6.8379', 'vk6.8390', 'vk6.8804', 'vk6.9296', 'vk6.9417', 'vk6.9744', 'vk6.17896', 'vk6.17961', 'vk6.18267', 'vk6.18604', 'vk6.24403', 'vk6.25159', 'vk6.30036', 'vk6.30097', 'vk6.36885', 'vk6.37345', 'vk6.39819', 'vk6.39841', 'vk6.43830', 'vk6.44106', 'vk6.44431', 'vk6.46383', 'vk6.46400', 'vk6.47960', 'vk6.47978', 'vk6.48628', 'vk6.49074', 'vk6.49911', 'vk6.50620', 'vk6.51142'] |
| 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 | O1O2O3O4O5U6U4O6U3U1U5U2 |
| R3 orbit | {'O1O2O3O4O5U6U4O6U3U1U5U2', 'O1O2O3O4U5U3O5U6U1U4O6U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U1U5U3O6U2U6 |
| Gauss code of K* | O1O2O3O4U2U4U1U5U3O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U2U6U4U1U3 |
| 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 1 -1 0 3 -1],[ 2 0 2 0 1 3 1],[-1 -2 0 -1 0 2 -2],[ 1 0 1 0 1 2 0],[ 0 -1 0 -1 0 0 0],[-3 -3 -2 -2 0 0 -3],[ 1 -1 2 0 0 3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -2 0 -2 -3 -3],[-1 2 0 0 -1 -2 -2],[ 0 0 0 0 -1 0 -1],[ 1 2 1 1 0 0 0],[ 1 3 2 0 0 0 -1],[ 2 3 2 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,2,0,2,3,3,0,1,2,2,1,0,1,0,0,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,3,1,2,2,1,0,1,1,1,0,1,0,0,1] |
| Phi of -K | [-2,-1,-1,0,1,3,0,1,1,1,2,0,1,0,1,0,1,2,1,3,0] |
| Phi of K* | [-3,-1,0,1,1,2,0,3,1,2,2,1,0,1,1,1,0,1,0,0,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,1,1,2,3,0,1,1,2,0,2,3,0,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+38t^4+48t^2+1 |
| Outer characteristic polynomial | t^7+54t^5+79t^3+4t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 256*K1**4*K2 - 1248*K1**4 + 96*K1**3*K2*K3 - 96*K1**3*K3 - 1264*K1**2*K2**2 - 64*K1**2*K2*K4 + 2904*K1**2*K2 - 224*K1**2*K3**2 - 32*K1**2*K3*K5 - 1896*K1**2 - 224*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 1952*K1*K2*K3 + 664*K1*K3*K4 + 152*K1*K4*K5 + 24*K1*K5*K6 - 88*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 416*K2**2*K4 - 1718*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 868*K3**2 - 446*K4**2 - 148*K5**2 - 18*K6**2 + 1844 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]] |
| If K is slice | False |