| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,3,-1,0,2,1,0,0,0,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1334'] |
| 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+33t^5+47t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1334'] |
| 2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1664*K1**4*K2 - 4160*K1**4 + 512*K1**3*K2*K3 - 512*K1**3*K3 - 512*K1**2*K2**4 + 2752*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11376*K1**2*K2**2 - 1056*K1**2*K2*K4 + 13064*K1**2*K2 - 128*K1**2*K3**2 - 64*K1**2*K4**2 - 6376*K1**2 + 768*K1*K2**3*K3 - 1856*K1*K2**2*K3 - 288*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9456*K1*K2*K3 + 984*K1*K3*K4 + 88*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2272*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 2256*K2**2*K4 - 4646*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 2028*K3**2 - 644*K4**2 - 44*K5**2 - 2*K6**2 + 5306 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1334'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3615', 'vk6.3688', 'vk6.3881', 'vk6.4000', 'vk6.7041', 'vk6.7084', 'vk6.7261', 'vk6.7374', 'vk6.17692', 'vk6.17741', 'vk6.24243', 'vk6.24304', 'vk6.36534', 'vk6.36611', 'vk6.43644', 'vk6.43751', 'vk6.48251', 'vk6.48328', 'vk6.48413', 'vk6.48432', 'vk6.50011', 'vk6.50054', 'vk6.50139', 'vk6.50156', 'vk6.55724', 'vk6.55781', 'vk6.60300', 'vk6.60383', 'vk6.65432', 'vk6.65461', 'vk6.68564', 'vk6.68593'] |
| 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 | O1O2O3U1O4O5U3U4U5O6U2U6 |
| R3 orbit | {'O1O2O3U1O4O5U3U4U5O6U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2O4U5U6U1O5O6U3 |
| Gauss code of K* | O1O2O3U4O5O4U6U5U1O6U2U3 |
| Gauss code of -K* | O1O2O3U1U2O4U3U5U4O6O5U6 |
| 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 -1 0 2 1],[ 2 0 2 1 1 1 1],[ 0 -2 0 -2 0 2 1],[ 1 -1 2 0 1 2 0],[ 0 -1 0 -1 0 1 0],[-2 -1 -2 -2 -1 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -2 -1],[-1 0 0 0 -1 0 -1],[ 0 1 0 0 0 -1 -1],[ 0 2 1 0 0 -2 -2],[ 1 2 0 1 2 0 -1],[ 2 1 1 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,2,2,1,0,1,0,1,0,1,1,2,2,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,2,3,-1,0,2,1,0,0,0,1,1,1] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,2,3,-1,0,2,1,0,0,0,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,0,1,1,3,0,1,2,2,0,-1,0,0,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,1,1,1,2,0,2,0,0,1,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+23t^4+23t^2 |
| Outer characteristic polynomial | t^7+33t^5+47t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -896*K1**4*K2**2 + 1664*K1**4*K2 - 4160*K1**4 + 512*K1**3*K2*K3 - 512*K1**3*K3 - 512*K1**2*K2**4 + 2752*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11376*K1**2*K2**2 - 1056*K1**2*K2*K4 + 13064*K1**2*K2 - 128*K1**2*K3**2 - 64*K1**2*K4**2 - 6376*K1**2 + 768*K1*K2**3*K3 - 1856*K1*K2**2*K3 - 288*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9456*K1*K2*K3 + 984*K1*K3*K4 + 88*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2272*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 2256*K2**2*K4 - 4646*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 2028*K3**2 - 644*K4**2 - 44*K5**2 - 2*K6**2 + 5306 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |