| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,3,3,1,0,1,2,1,0,1,1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1863', '7.38519'] |
| 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+39t^5+123t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1863'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 512*K1**4*K2 - 960*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 256*K1**2*K2**3 - 1488*K1**2*K2**2 + 1880*K1**2*K2 - 128*K1**2*K3**2 - 48*K1**2*K4**2 - 1040*K1**2 + 64*K1*K2**3*K3 + 1224*K1*K2*K3 + 248*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 304*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 256*K2**2*K4 - 846*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 424*K3**2 - 188*K4**2 - 24*K5**2 - 2*K6**2 + 1082 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1863'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11277', 'vk6.11357', 'vk6.12542', 'vk6.12655', 'vk6.18356', 'vk6.18695', 'vk6.24804', 'vk6.25261', 'vk6.30953', 'vk6.31078', 'vk6.32133', 'vk6.32254', 'vk6.36990', 'vk6.37441', 'vk6.44173', 'vk6.44493', 'vk6.52045', 'vk6.52130', 'vk6.52888', 'vk6.52953', 'vk6.56140', 'vk6.56367', 'vk6.60665', 'vk6.61010', 'vk6.63658', 'vk6.63705', 'vk6.64090', 'vk6.64137', 'vk6.65804', 'vk6.66057', 'vk6.68803', 'vk6.69012'] |
| 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 | O1O2O3U4U5O4U2O6O5U1U6U3 |
| R3 orbit | {'O1O2O3U4U5O4U2O6O5U1U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U3O5O4U2O6U5U6 |
| Gauss code of K* | O1O2O3U1U4U3O5O6U5O4U2U6 |
| Gauss code of -K* | O1O2O3U4U2O5U6O4O6U1U5U3 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 0 2 -1 1 0],[ 2 0 1 3 1 2 0],[ 0 -1 0 0 0 1 -1],[-2 -3 0 0 -3 0 -1],[ 1 -1 0 3 0 1 1],[-1 -2 -1 0 -1 0 0],[ 0 0 1 1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -3 -3],[-1 0 0 -1 0 -1 -2],[ 0 0 1 0 -1 0 -1],[ 0 1 0 1 0 -1 0],[ 1 3 1 0 1 0 -1],[ 2 3 2 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,1,3,3,1,0,1,2,1,0,1,1,0,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,3,3,1,0,1,2,1,0,1,1,0,1] |
| Phi of -K | [-2,-1,0,0,1,2,0,1,2,1,1,1,0,1,0,1,0,2,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,2,0,1,1,0,1,1,1,0,2,1,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,0,1,2,3,1,0,1,3,1,0,1,1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 9z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+11w^2z+19w |
| Inner characteristic polynomial | t^6+29t^4+87t^2 |
| Outer characteristic polynomial | t^7+39t^5+123t^3 |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 512*K1**4*K2 - 960*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 256*K1**2*K2**3 - 1488*K1**2*K2**2 + 1880*K1**2*K2 - 128*K1**2*K3**2 - 48*K1**2*K4**2 - 1040*K1**2 + 64*K1*K2**3*K3 + 1224*K1*K2*K3 + 248*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 304*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 256*K2**2*K4 - 846*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 424*K3**2 - 188*K4**2 - 24*K5**2 - 2*K6**2 + 1082 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | True |