| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,2,3,1,1,1,2,1,1,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1056'] |
| 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+40t^5+57t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1056'] |
| 2-strand cable arrow polynomial of the knot is: 640*K1**4*K2 - 2144*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 960*K1**3*K3 - 2720*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 7384*K1**2*K2 - 1120*K1**2*K3**2 - 32*K1**2*K3*K5 - 112*K1**2*K4**2 - 6204*K1**2 - 704*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 6648*K1*K2*K3 + 1808*K1*K3*K4 + 112*K1*K4*K5 - 248*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 840*K2**2*K4 - 4686*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 2596*K3**2 - 762*K4**2 - 48*K5**2 - 2*K6**2 + 4856 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1056'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81567', 'vk6.81580', 'vk6.81649', 'vk6.81660', 'vk6.81733', 'vk6.81734', 'vk6.81863', 'vk6.81864', 'vk6.82231', 'vk6.82248', 'vk6.82389', 'vk6.82390', 'vk6.82504', 'vk6.82505', 'vk6.82579', 'vk6.82580', 'vk6.83167', 'vk6.83175', 'vk6.83594', 'vk6.83605', 'vk6.84133', 'vk6.84144', 'vk6.84342', 'vk6.84343', 'vk6.84567', 'vk6.84568', 'vk6.86478', 'vk6.86494', 'vk6.88726', 'vk6.88727', 'vk6.88922', 'vk6.88924'] |
| 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 | O1O2O3O4U5U1O6U2O5U3U6U4 |
| R3 orbit | {'O1O2O3O4U5U1O6U2O5U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U2O6U3O5U4U6 |
| Gauss code of K* | O1O2O3U4U5U1U3O6O4U2O5U6 |
| Gauss code of -K* | O1O2O3U4O5U2O6O4U1U3U5U6 |
| 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 0 3 -1 1],[ 2 0 1 1 2 1 1],[ 1 -1 0 0 2 1 1],[ 0 -1 0 0 2 0 0],[-3 -2 -2 -2 0 -2 -1],[ 1 -1 -1 0 2 0 1],[-1 -1 -1 0 1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -2 -2 -2],[-1 1 0 0 -1 -1 -1],[ 0 2 0 0 0 0 -1],[ 1 2 1 0 0 1 -1],[ 1 2 1 0 -1 0 -1],[ 2 2 1 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,2,2,2,2,0,1,1,1,0,0,1,-1,1,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,2,2,3,1,1,1,2,1,1,1,-1,0,0] |
| Phi of -K | [-2,-1,-1,0,1,3,0,0,1,2,3,-1,1,1,2,1,1,2,1,1,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,1,2,2,3,1,1,1,2,1,1,1,-1,0,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,1,1,1,2,-1,0,1,2,0,1,2,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+24t^4+24t^2+1 |
| Outer characteristic polynomial | t^7+40t^5+57t^3+8t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | 640*K1**4*K2 - 2144*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 960*K1**3*K3 - 2720*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 7384*K1**2*K2 - 1120*K1**2*K3**2 - 32*K1**2*K3*K5 - 112*K1**2*K4**2 - 6204*K1**2 - 704*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 6648*K1*K2*K3 + 1808*K1*K3*K4 + 112*K1*K4*K5 - 248*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 840*K2**2*K4 - 4686*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 2596*K3**2 - 762*K4**2 - 48*K5**2 - 2*K6**2 + 4856 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}]] |
| If K is slice | False |