| Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,-1,1,2,1,4,1,1,0,1,-1,0,1,0,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.608'] |
| Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923'] |
| Outer characteristic polynomial of the knot is: t^7+52t^5+81t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.608'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1376*K1**4*K2 - 3712*K1**4 + 384*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 + 1952*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 8432*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 10688*K1**2*K2 - 1504*K1**2*K3**2 - 32*K1**2*K3*K5 - 5944*K1**2 + 608*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 64*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9256*K1*K2*K3 + 1760*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1824*K2**4 - 976*K2**2*K3**2 - 80*K2**2*K4**2 + 1768*K2**2*K4 - 4416*K2**2 + 672*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 64*K3**2*K6 - 2344*K3**2 - 608*K4**2 - 80*K5**2 - 16*K6**2 + 5102 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.608'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4410', 'vk6.4505', 'vk6.5792', 'vk6.5919', 'vk6.7863', 'vk6.7968', 'vk6.9279', 'vk6.9398', 'vk6.10172', 'vk6.10245', 'vk6.10390', 'vk6.17873', 'vk6.17936', 'vk6.18295', 'vk6.18632', 'vk6.24380', 'vk6.25186', 'vk6.30059', 'vk6.30122', 'vk6.36905', 'vk6.37365', 'vk6.43807', 'vk6.44126', 'vk6.44451', 'vk6.48617', 'vk6.50522', 'vk6.50605', 'vk6.51125', 'vk6.51673', 'vk6.55840', 'vk6.56081', 'vk6.65502'] |
| 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 | O1O2O3O4U3O5U1O6U5U6U2U4 |
| R3 orbit | {'O1O2O3O4U3O5U1O6U5U6U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U3U5U6O5U4O6U2 |
| Gauss code of K* | O1O2O3O4U5U3U6U4O6U1O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5U4O6U1U6U2U5 |
| 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 -3 0 -1 3 0 1],[ 3 0 2 0 4 1 1],[ 0 -2 0 0 2 -1 1],[ 1 0 0 0 1 0 0],[-3 -4 -2 -1 0 -1 1],[ 0 -1 1 0 1 0 1],[-1 -1 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 -1 -3],[-3 0 1 -1 -2 -1 -4],[-1 -1 0 -1 -1 0 -1],[ 0 1 1 0 1 0 -1],[ 0 2 1 -1 0 0 -2],[ 1 1 0 0 0 0 0],[ 3 4 1 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,1,3,-1,1,2,1,4,1,1,0,1,-1,0,1,0,2,0] |
| Phi over symmetry | [-3,-1,0,0,1,3,-1,1,2,1,4,1,1,0,1,-1,0,1,0,2,0] |
| Phi of -K | [-3,-1,0,0,1,3,2,1,2,3,2,1,1,2,3,1,0,1,0,2,3] |
| Phi of K* | [-3,-1,0,0,1,3,3,1,2,3,2,0,0,2,3,-1,1,1,1,2,2] |
| Phi of -K* | [-3,-1,0,0,1,3,0,1,2,1,4,0,0,0,1,1,1,1,1,2,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+24z+37 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+24w^2z+37w |
| Inner characteristic polynomial | t^6+32t^4+33t^2 |
| Outer characteristic polynomial | t^7+52t^5+81t^3+7t |
| Flat arrow polynomial | 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 1376*K1**4*K2 - 3712*K1**4 + 384*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 + 1952*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 8432*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 10688*K1**2*K2 - 1504*K1**2*K3**2 - 32*K1**2*K3*K5 - 5944*K1**2 + 608*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 64*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9256*K1*K2*K3 + 1760*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1824*K2**4 - 976*K2**2*K3**2 - 80*K2**2*K4**2 + 1768*K2**2*K4 - 4416*K2**2 + 672*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 64*K3**2*K6 - 2344*K3**2 - 608*K4**2 - 80*K5**2 - 16*K6**2 + 5102 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |