| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,2,2,2,1,1,1,1,1,0,0,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1034'] |
| Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863'] |
| Outer characteristic polynomial of the knot is: t^7+37t^5+68t^3+14t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1034'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1568*K1**4*K2 - 3536*K1**4 - 896*K1**3*K3 + 832*K1**2*K2**3 - 5280*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 10752*K1**2*K2 - 208*K1**2*K3**2 - 6900*K1**2 + 128*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6176*K1*K2*K3 + 616*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1120*K2**4 - 304*K2**2*K3**2 - 48*K2**2*K4**2 + 1240*K2**2*K4 - 4878*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 1784*K3**2 - 400*K4**2 - 28*K5**2 - 2*K6**2 + 5158 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1034'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17104', 'vk6.17346', 'vk6.20571', 'vk6.21980', 'vk6.23497', 'vk6.23834', 'vk6.28037', 'vk6.29496', 'vk6.35646', 'vk6.36084', 'vk6.39439', 'vk6.41640', 'vk6.43011', 'vk6.43322', 'vk6.46027', 'vk6.47695', 'vk6.55255', 'vk6.55506', 'vk6.57453', 'vk6.58620', 'vk6.59663', 'vk6.60009', 'vk6.62128', 'vk6.63094', 'vk6.65063', 'vk6.65255', 'vk6.66981', 'vk6.67846', 'vk6.68325', 'vk6.68474', 'vk6.69600', 'vk6.70293'] |
| 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 | O1O2O3O4U5U1O5U3O6U4U2U6 |
| R3 orbit | {'O1O2O3O4U5U1O5U3O6U4U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U1O5U2O6U4U6 |
| Gauss code of K* | O1O2O3U4U2U5U1O6O4U6O5U3 |
| Gauss code of -K* | O1O2O3U1O4U5O6O5U3U4U2U6 |
| 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 0 0 1 -1 2],[ 2 0 2 0 1 2 2],[ 0 -2 0 -1 1 0 2],[ 0 0 1 0 1 0 1],[-1 -1 -1 -1 0 -1 1],[ 1 -2 0 0 1 0 2],[-2 -2 -2 -1 -1 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -2 -2],[-1 1 0 -1 -1 -1 -1],[ 0 1 1 0 1 0 0],[ 0 2 1 -1 0 0 -2],[ 1 2 1 0 0 0 -2],[ 2 2 1 0 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,1,2,2,2,1,1,1,1,-1,0,0,0,2,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,0,2,2,2,1,1,1,1,1,0,0,0,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,-1,0,2,2,2,1,1,1,1,1,0,0,0,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,0,1,1,2,0,0,1,2,-1,1,0,1,2,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,0,2,1,2,0,0,1,2,1,1,1,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+27t^4+28t^2+4 |
| Outer characteristic polynomial | t^7+37t^5+68t^3+14t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 1568*K1**4*K2 - 3536*K1**4 - 896*K1**3*K3 + 832*K1**2*K2**3 - 5280*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 10752*K1**2*K2 - 208*K1**2*K3**2 - 6900*K1**2 + 128*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6176*K1*K2*K3 + 616*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1120*K2**4 - 304*K2**2*K3**2 - 48*K2**2*K4**2 + 1240*K2**2*K4 - 4878*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 1784*K3**2 - 400*K4**2 - 28*K5**2 - 2*K6**2 + 5158 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{6}, {3, 5}, {1, 4}, {2}]] |
| If K is slice | False |