| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,0,0,1,1,-1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1822', '7.42380'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+54t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1822'] |
| 2-strand cable arrow polynomial of the knot is: -1664*K1**4*K2**2 + 3232*K1**4*K2 - 4672*K1**4 + 1248*K1**3*K2*K3 - 576*K1**3*K3 - 1536*K1**2*K2**4 + 5440*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12160*K1**2*K2**2 - 896*K1**2*K2*K4 + 8136*K1**2*K2 - 160*K1**2*K3**2 - 940*K1**2 + 2144*K1*K2**3*K3 - 2784*K1*K2**2*K3 - 384*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6368*K1*K2*K3 + 320*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3640*K2**4 - 32*K2**3*K6 - 720*K2**2*K3**2 - 16*K2**2*K4**2 + 2328*K2**2*K4 - 390*K2**2 + 232*K2*K3*K5 + 16*K2*K4*K6 - 656*K3**2 - 178*K4**2 - 20*K5**2 - 2*K6**2 + 1880 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1822'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.478', 'vk6.544', 'vk6.581', 'vk6.946', 'vk6.1044', 'vk6.1083', 'vk6.1631', 'vk6.1736', 'vk6.1815', 'vk6.2124', 'vk6.2227', 'vk6.2267', 'vk6.2549', 'vk6.2868', 'vk6.3042', 'vk6.3172', 'vk6.20415', 'vk6.20711', 'vk6.21778', 'vk6.22154', 'vk6.27768', 'vk6.28260', 'vk6.29291', 'vk6.29684', 'vk6.39192', 'vk6.39720', 'vk6.41968', 'vk6.46283', 'vk6.57283', 'vk6.57641', 'vk6.58532', 'vk6.61951'] |
| 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 | O1O2O3U1U2O4U3O5O6U5U6U4 |
| R3 orbit | {'O1O2O3U1U2O4U3O5O6U5U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U6O5O6U1O4U2U3 |
| Gauss code of K* | O1O2O3U4U5U6O4O5U3O6U1U2 |
| Gauss code of -K* | O1O2O3U2U3O4U1O5O6U4U5U6 |
| 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 1 -1 1],[ 2 0 1 2 1 0 0],[ 0 -1 0 1 1 0 0],[-1 -2 -1 0 1 0 0],[-1 -1 -1 -1 0 -1 1],[ 1 0 0 0 1 0 1],[-1 0 0 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 0 -2],[-1 -1 0 1 -1 -1 -1],[-1 0 -1 0 0 -1 0],[ 0 1 1 0 0 0 -1],[ 1 0 1 1 0 0 0],[ 2 2 1 0 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,0,2,-1,1,1,1,0,1,0,0,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,0,0,1,1,-1,0,-1] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,1,2,3,1,2,1,1,0,0,1,-1,0,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,3,-1,0,1,2,0,2,1,1,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,0,0,1,1,-1,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+25z+27 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+25w^2z+27w |
| Inner characteristic polynomial | t^6+12t^4+21t^2 |
| Outer characteristic polynomial | t^7+20t^5+54t^3+5t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -1664*K1**4*K2**2 + 3232*K1**4*K2 - 4672*K1**4 + 1248*K1**3*K2*K3 - 576*K1**3*K3 - 1536*K1**2*K2**4 + 5440*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 12160*K1**2*K2**2 - 896*K1**2*K2*K4 + 8136*K1**2*K2 - 160*K1**2*K3**2 - 940*K1**2 + 2144*K1*K2**3*K3 - 2784*K1*K2**2*K3 - 384*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6368*K1*K2*K3 + 320*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3640*K2**4 - 32*K2**3*K6 - 720*K2**2*K3**2 - 16*K2**2*K4**2 + 2328*K2**2*K4 - 390*K2**2 + 232*K2*K3*K5 + 16*K2*K4*K6 - 656*K3**2 - 178*K4**2 - 20*K5**2 - 2*K6**2 + 1880 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {3, 5}, {2, 4}, {1}]] |
| If K is slice | False |