| Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,1,2,3,0,1,2,2,0,0,0,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.945'] |
| Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376'] |
| Outer characteristic polynomial of the knot is: t^7+53t^5+61t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.945'] |
| 2-strand cable arrow polynomial of the knot is: -784*K1**4 + 128*K1**3*K2*K3 + 32*K1**3*K3*K4 - 192*K1**3*K3 + 352*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2992*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 4304*K1**2*K2 - 176*K1**2*K3**2 - 48*K1**2*K4**2 - 2900*K1**2 + 512*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 576*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3568*K1*K2*K3 - 32*K1*K2*K4*K5 + 480*K1*K3*K4 + 40*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 896*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 40*K2**2*K4**2 + 832*K2**2*K4 - 1790*K2**2 + 224*K2*K3*K5 + 24*K2*K4*K6 - 928*K3**2 - 212*K4**2 - 20*K5**2 - 2*K6**2 + 2082 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.945'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71355', 'vk6.71407', 'vk6.71417', 'vk6.71872', 'vk6.71884', 'vk6.71933', 'vk6.71943', 'vk6.74192', 'vk6.74320', 'vk6.74335', 'vk6.74965', 'vk6.74981', 'vk6.75621', 'vk6.75811', 'vk6.76355', 'vk6.76536', 'vk6.76551', 'vk6.76943', 'vk6.77011', 'vk6.77015', 'vk6.77070', 'vk6.78604', 'vk6.78804', 'vk6.79225', 'vk6.79366', 'vk6.79790', 'vk6.79806', 'vk6.80246', 'vk6.80704', 'vk6.80839', 'vk6.81273', 'vk6.81487', 'vk6.84055', 'vk6.86002', 'vk6.87079', 'vk6.87083', 'vk6.87742', 'vk6.88036', 'vk6.88213', 'vk6.89400'] |
| 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 | O1O2O3O4U5U1O6O5U2U3U6U4 |
| R3 orbit | {'O1O2O3O4U5U1O6O5U2U3U6U4', 'O1O2O3O4U2U5O6U1O5U3U6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U1U5U2U3O6O5U4U6 |
| Gauss code of K* | O1O2O3O4U5U1U2U4O6O5U3U6 |
| Gauss code of -K* | O1O2O3O4U5U2O6O5U1U3U4U6 |
| 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 -2 0 3 0 1],[ 2 0 0 1 2 2 1],[ 2 0 0 1 3 2 1],[ 0 -1 -1 0 2 0 0],[-3 -2 -3 -2 0 -2 -1],[ 0 -2 -2 0 2 0 1],[-1 -1 -1 0 1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 -2 -2],[-3 0 -1 -2 -2 -2 -3],[-1 1 0 0 -1 -1 -1],[ 0 2 0 0 0 -1 -1],[ 0 2 1 0 0 -2 -2],[ 2 2 1 1 2 0 0],[ 2 3 1 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,2,2,1,2,2,2,3,0,1,1,1,0,1,1,2,2,0] |
| Phi over symmetry | [-3,-1,0,0,2,2,1,1,1,2,3,0,1,2,2,0,0,0,1,1,0] |
| Phi of -K | [-2,-2,0,0,1,3,0,0,1,2,2,0,1,2,3,0,0,1,1,1,1] |
| Phi of K* | [-3,-1,0,0,2,2,1,1,1,2,3,0,1,2,2,0,0,0,1,1,0] |
| Phi of -K* | [-2,-2,0,0,1,3,0,1,2,1,2,1,2,1,3,0,0,2,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+15w^2z+23w |
| Inner characteristic polynomial | t^6+35t^4+35t^2+1 |
| Outer characteristic polynomial | t^7+53t^5+61t^3+4t |
| Flat arrow polynomial | 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -784*K1**4 + 128*K1**3*K2*K3 + 32*K1**3*K3*K4 - 192*K1**3*K3 + 352*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2992*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 4304*K1**2*K2 - 176*K1**2*K3**2 - 48*K1**2*K4**2 - 2900*K1**2 + 512*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 576*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3568*K1*K2*K3 - 32*K1*K2*K4*K5 + 480*K1*K3*K4 + 40*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 896*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 40*K2**2*K4**2 + 832*K2**2*K4 - 1790*K2**2 + 224*K2*K3*K5 + 24*K2*K4*K6 - 928*K3**2 - 212*K4**2 - 20*K5**2 - 2*K6**2 + 2082 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |