| Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,0,0,2,2,4,0,1,0,2,1,-1,0,0,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.580'] |
| Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.206', '6.236', '6.575', '6.580', '6.613', '6.619', '6.810', '6.819', '6.831', '6.838', '6.957', '6.1018', '6.1028', '6.1046', '6.1073', '6.1279', '6.1507', '6.1532', '6.1556', '6.1639', '6.1688', '6.1924', '6.1931'] |
| Outer characteristic polynomial of the knot is: t^7+68t^5+87t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.580'] |
| 2-strand cable arrow polynomial of the knot is: 224*K1**4*K2 - 608*K1**4 - 96*K1**3*K3 + 160*K1**2*K2**3 - 3408*K1**2*K2**2 - 32*K1**2*K2*K4 + 5384*K1**2*K2 - 4376*K1**2 + 128*K1*K2**3*K3 - 160*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3968*K1*K2*K3 + 176*K1*K3*K4 + 8*K1*K4*K5 - 320*K2**6 + 192*K2**4*K4 - 2480*K2**4 - 336*K2**2*K3**2 - 64*K2**2*K4**2 + 2080*K2**2*K4 - 2276*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 - 1188*K3**2 - 452*K4**2 - 44*K5**2 - 4*K6**2 + 3290 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.580'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73773', 'vk6.73776', 'vk6.73796', 'vk6.73801', 'vk6.73911', 'vk6.73913', 'vk6.73930', 'vk6.73937', 'vk6.75739', 'vk6.75744', 'vk6.75911', 'vk6.75912', 'vk6.78706', 'vk6.78715', 'vk6.78750', 'vk6.78753', 'vk6.78905', 'vk6.78917', 'vk6.80327', 'vk6.80334', 'vk6.80352', 'vk6.80354', 'vk6.80450', 'vk6.80456', 'vk6.81713', 'vk6.81722', 'vk6.82488', 'vk6.82494', 'vk6.84452', 'vk6.84460', 'vk6.88343', 'vk6.88354'] |
| 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 | O1O2O3O4U1O5U3O6U5U4U2U6 |
| R3 orbit | {'O1O2O3O4U1O5U3O6U5U4U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U1U6O5U2O6U4 |
| Gauss code of K* | O1O2O3O4U5U3U6U2O5U1O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5U4O6U3U5U2U6 |
| 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 1 0 3],[ 3 0 3 1 2 1 2],[ 0 -3 0 -2 1 1 3],[ 1 -1 2 0 2 1 2],[-1 -2 -1 -2 0 0 2],[ 0 -1 -1 -1 0 0 1],[-3 -2 -3 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 -1 -3],[-3 0 -2 -1 -3 -2 -2],[-1 2 0 0 -1 -2 -2],[ 0 1 0 0 -1 -1 -1],[ 0 3 1 1 0 -2 -3],[ 1 2 2 1 2 0 -1],[ 3 2 2 1 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,1,3,2,1,3,2,2,0,1,2,2,1,1,1,2,3,1] |
| Phi over symmetry | [-3,-1,0,0,1,3,0,0,2,2,4,0,1,0,2,1,-1,0,0,2,1] |
| Phi of -K | [-3,-1,0,0,1,3,1,0,2,2,4,-1,0,0,2,-1,0,0,1,2,0] |
| Phi of K* | [-3,-1,0,0,1,3,0,0,2,2,4,0,1,0,2,1,-1,0,0,2,1] |
| Phi of -K* | [-3,-1,0,0,1,3,1,1,3,2,2,1,2,2,2,-1,0,1,1,3,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+4w^3z^2-8w^3z+23w^2z+23w |
| Inner characteristic polynomial | t^6+48t^4+19t^2+1 |
| Outer characteristic polynomial | t^7+68t^5+87t^3+16t |
| Flat arrow polynomial | 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| 2-strand cable arrow polynomial | 224*K1**4*K2 - 608*K1**4 - 96*K1**3*K3 + 160*K1**2*K2**3 - 3408*K1**2*K2**2 - 32*K1**2*K2*K4 + 5384*K1**2*K2 - 4376*K1**2 + 128*K1*K2**3*K3 - 160*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3968*K1*K2*K3 + 176*K1*K3*K4 + 8*K1*K4*K5 - 320*K2**6 + 192*K2**4*K4 - 2480*K2**4 - 336*K2**2*K3**2 - 64*K2**2*K4**2 + 2080*K2**2*K4 - 2276*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 - 1188*K3**2 - 452*K4**2 - 44*K5**2 - 4*K6**2 + 3290 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}]] |
| If K is slice | False |