| Min(phi) over symmetries of the knot is: [-3,-3,-1,1,3,3,-1,1,1,4,5,1,0,2,3,0,1,2,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.310'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+101t^5+136t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.310'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4 + 448*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 2688*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 3504*K1**2*K2 - 480*K1**2*K3**2 - 3064*K1**2 + 1024*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 4416*K1*K2*K3 + 744*K1*K3*K4 - 880*K2**4 - 1024*K2**2*K3**2 - 80*K2**2*K4**2 + 864*K2**2*K4 - 1980*K2**2 - 64*K2*K3**2*K4 + 408*K2*K3*K5 + 40*K2*K4*K6 - 32*K3**4 + 40*K3**2*K6 - 1464*K3**2 - 332*K4**2 - 40*K5**2 - 20*K6**2 + 2386 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.310'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11508', 'vk6.11831', 'vk6.12848', 'vk6.13165', 'vk6.20273', 'vk6.21596', 'vk6.27539', 'vk6.29109', 'vk6.31273', 'vk6.31647', 'vk6.32421', 'vk6.32846', 'vk6.38944', 'vk6.41177', 'vk6.45711', 'vk6.47418', 'vk6.52271', 'vk6.52526', 'vk6.53102', 'vk6.53428', 'vk6.57110', 'vk6.58290', 'vk6.61695', 'vk6.62846', 'vk6.63790', 'vk6.63912', 'vk6.64228', 'vk6.64434', 'vk6.66741', 'vk6.67619', 'vk6.69395', 'vk6.70123'] |
| 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 | O1O2O3O4O5U2U3O6U1U6U5U4 |
| R3 orbit | {'O1O2O3O4O5U2U3O6U1U6U5U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U1U6U5O6U3U4 |
| Gauss code of K* | O1O2O3O4U1U5U6U4U3O5O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U2U1U5U6U4 |
| 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 -3 -1 3 3 1],[ 3 0 -1 1 5 4 1],[ 3 1 0 1 3 2 0],[ 1 -1 -1 0 2 1 0],[-3 -5 -3 -2 0 0 0],[-3 -4 -2 -1 0 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 3 1 -1 -3 -3],[-3 0 0 0 -1 -2 -4],[-3 0 0 0 -2 -3 -5],[-1 0 0 0 0 0 -1],[ 1 1 2 0 0 -1 -1],[ 3 2 3 0 1 0 1],[ 3 4 5 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-3,-1,1,3,3,0,0,1,2,4,0,2,3,5,0,0,1,1,1,-1] |
| Phi over symmetry | [-3,-3,-1,1,3,3,-1,1,1,4,5,1,0,2,3,0,1,2,0,0,0] |
| Phi of -K | [-3,-3,-1,1,3,3,-1,1,4,3,4,1,3,1,2,2,2,3,2,2,0] |
| Phi of K* | [-3,-3,-1,1,3,3,0,2,2,1,3,2,3,2,4,2,3,4,1,1,-1] |
| Phi of -K* | [-3,-3,-1,1,3,3,-1,1,1,4,5,1,0,2,3,0,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+23w^2z+23w |
| Inner characteristic polynomial | t^6+63t^4+34t^2+1 |
| Outer characteristic polynomial | t^7+101t^5+136t^3+5t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -320*K1**4 + 448*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 2688*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 3504*K1**2*K2 - 480*K1**2*K3**2 - 3064*K1**2 + 1024*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 4416*K1*K2*K3 + 744*K1*K3*K4 - 880*K2**4 - 1024*K2**2*K3**2 - 80*K2**2*K4**2 + 864*K2**2*K4 - 1980*K2**2 - 64*K2*K3**2*K4 + 408*K2*K3*K5 + 40*K2*K4*K6 - 32*K3**4 + 40*K3**2*K6 - 1464*K3**2 - 332*K4**2 - 40*K5**2 - 20*K6**2 + 2386 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | True |