| Min(phi) over symmetries of the knot is: [-4,-4,1,1,3,3,0,1,3,2,4,2,4,3,5,1,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.65'] |
| 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+142t^5+293t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.65'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 - 256*K1**2*K2**4 + 448*K1**2*K2**3 - 2016*K1**2*K2**2 + 1104*K1**2*K2 - 128*K1**2*K3**2 - 1344*K1**2 + 1152*K1*K2**3*K3 + 2832*K1*K2*K3 + 176*K1*K3*K4 + 16*K1*K4*K5 - 512*K2**8 + 256*K2**6*K4 - 768*K2**6 - 256*K2**4*K3**2 + 512*K2**4*K4 - 1072*K2**4 + 320*K2**3*K3*K5 - 1376*K2**2*K3**2 - 16*K2**2*K4**2 + 464*K2**2*K4 - 128*K2**2*K5**2 - 76*K2**2 + 656*K2*K3*K5 + 16*K2*K4*K6 + 16*K2*K5*K7 - 64*K3**4 + 32*K3**2*K6 - 1000*K3**2 - 76*K4**2 - 120*K5**2 - 4*K6**2 + 1274 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.65'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81109', 'vk6.81130', 'vk6.81134', 'vk6.81228', 'vk6.81231', 'vk6.81249', 'vk6.81250', 'vk6.82105', 'vk6.82629', 'vk6.82890', 'vk6.83408', 'vk6.83413', 'vk6.84008', 'vk6.86332', 'vk6.86334', 'vk6.87905', 'vk6.88565', 'vk6.89066', 'vk6.89069', 'vk6.90047'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is +. |
| The reverse -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 | O1O2O3O4O5O6U2U1U5U6U3U4 |
| R3 orbit | {'O1O2O3O4O5O6U2U1U5U6U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5O6U3U4U1U2U6U5 |
| Gauss code of K* | Same |
| Gauss code of -K* | O1O2O3O4O5O6U3U4U1U2U6U5 |
| Diagrammatic symmetry type | + |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -4 -4 1 3 1 3],[ 4 0 0 4 5 2 3],[ 4 0 0 3 4 1 2],[-1 -4 -3 0 1 -1 1],[-3 -5 -4 -1 0 -1 1],[-1 -2 -1 1 1 0 1],[-3 -3 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 3 1 1 -4 -4],[-3 0 1 -1 -1 -4 -5],[-3 -1 0 -1 -1 -2 -3],[-1 1 1 0 1 -1 -2],[-1 1 1 -1 0 -3 -4],[ 4 4 2 1 3 0 0],[ 4 5 3 2 4 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-3,-1,-1,4,4,-1,1,1,4,5,1,1,2,3,-1,1,2,3,4,0] |
| Phi over symmetry | [-4,-4,1,1,3,3,0,1,3,2,4,2,4,3,5,1,1,1,1,1,-1] |
| Phi of -K | [-4,-4,1,1,3,3,0,1,3,2,4,2,4,3,5,1,1,1,1,1,-1] |
| Phi of K* | [-3,-3,-1,-1,4,4,-1,1,1,4,5,1,1,2,3,-1,1,2,3,4,0] |
| Phi of -K* | [-4,-4,1,1,3,3,0,1,3,2,4,2,4,3,5,1,1,1,1,1,-1] |
| Symmetry type of based matrix | + |
| u-polynomial | 2t^4-2t^3-2t |
| Normalized Jones-Krushkal polynomial | 4z+9 |
| Enhanced Jones-Krushkal polynomial | -8w^5z+16w^4z-10w^3z+4w^3+6w^2z+5w |
| Inner characteristic polynomial | t^6+90t^4+41t^2 |
| Outer characteristic polynomial | t^7+142t^5+293t^3 |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -64*K1**4 - 256*K1**2*K2**4 + 448*K1**2*K2**3 - 2016*K1**2*K2**2 + 1104*K1**2*K2 - 128*K1**2*K3**2 - 1344*K1**2 + 1152*K1*K2**3*K3 + 2832*K1*K2*K3 + 176*K1*K3*K4 + 16*K1*K4*K5 - 512*K2**8 + 256*K2**6*K4 - 768*K2**6 - 256*K2**4*K3**2 + 512*K2**4*K4 - 1072*K2**4 + 320*K2**3*K3*K5 - 1376*K2**2*K3**2 - 16*K2**2*K4**2 + 464*K2**2*K4 - 128*K2**2*K5**2 - 76*K2**2 + 656*K2*K3*K5 + 16*K2*K4*K6 + 16*K2*K5*K7 - 64*K3**4 + 32*K3**2*K6 - 1000*K3**2 - 76*K4**2 - 120*K5**2 - 4*K6**2 + 1274 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |