| Min(phi) over symmetries of the knot is: [-2,-1,1,1,1,0,0,1,2,1,0,1,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['5.82', '6.1734', '7.13522', '7.32803'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^6+30t^4+36t^2+1 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1734', '7.9995', '7.15891'] |
| 2-strand cable arrow polynomial of the knot is: 4608*K1**4*K2 - 9952*K1**4 + 2304*K1**3*K2*K3 - 1088*K1**3*K3 - 384*K1**2*K2**4 + 768*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9488*K1**2*K2**2 - 800*K1**2*K2*K4 + 8936*K1**2*K2 - 1632*K1**2*K3**2 - 96*K1**2*K4**2 + 1872*K1**2 + 512*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 256*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5472*K1*K2*K3 + 1112*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 696*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 680*K2**2*K4 - 1478*K2**2 + 168*K2*K3*K5 + 16*K2*K4*K6 - 684*K3**2 - 154*K4**2 - 20*K5**2 - 2*K6**2 + 1648 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1734'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3214', 'vk6.3224', 'vk6.3229', 'vk6.3234', 'vk6.3241', 'vk6.3321', 'vk6.3332', 'vk6.3336', 'vk6.3345', 'vk6.3350', 'vk6.3357', 'vk6.3446', 'vk6.3448', 'vk6.3503', 'vk6.15213', 'vk6.15222', 'vk6.15223', 'vk6.15236', 'vk6.15246', 'vk6.15255', 'vk6.33871', 'vk6.33887', 'vk6.33893', 'vk6.33894', 'vk6.33896', 'vk6.34331', 'vk6.34334', 'vk6.48089', 'vk6.48095', 'vk6.48154', 'vk6.48162', 'vk6.54441'] |
| 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 | O1O2O3U2U4O5O4U6U5O6U1U3 |
| R3 orbit | {'O1O2O3U2U4O5O4U6U5O6U1U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U3O4U5U4O6O5U6U2 |
| Gauss code of K* | O1O2U1O3O4U3U5U4O5O6U2U6 |
| Gauss code of -K* | O1O2U3O4O3U5U4O5O6U1U6U2 |
| 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 -1 -1 2 1 0 -1],[ 1 0 0 2 2 0 0],[ 1 0 0 1 1 0 1],[-2 -2 -1 0 -1 0 -3],[-1 -2 -1 1 0 0 -1],[ 0 0 0 0 0 0 0],[ 1 0 -1 3 1 0 0]] |
| Primitive based matrix | [[ 0 2 1 -1 -1 -1],[-2 0 -1 -1 -2 -3],[-1 1 0 -1 -2 -1],[ 1 1 1 0 0 1],[ 1 2 2 0 0 0],[ 1 3 1 -1 0 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-2,-1,1,1,1,1,1,2,3,1,2,1,0,-1,0] |
| Phi over symmetry | [-2,-1,1,1,1,0,0,1,2,1,0,1,0,-1,0] |
| Phi of -K | [-1,-1,-1,1,2,-1,0,1,2,0,1,0,0,1,0] |
| Phi of K* | [-2,-1,1,1,1,0,0,1,2,1,0,1,0,-1,0] |
| Phi of -K* | [-1,-1,-1,1,2,-1,0,1,3,0,1,1,2,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 9z^2+30z+25 |
| Enhanced Jones-Krushkal polynomial | 9w^3z^2+30w^2z+25w |
| Inner characteristic polynomial | t^5+22t^3+25t |
| Outer characteristic polynomial | t^6+30t^4+36t^2+1 |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 4608*K1**4*K2 - 9952*K1**4 + 2304*K1**3*K2*K3 - 1088*K1**3*K3 - 384*K1**2*K2**4 + 768*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9488*K1**2*K2**2 - 800*K1**2*K2*K4 + 8936*K1**2*K2 - 1632*K1**2*K3**2 - 96*K1**2*K4**2 + 1872*K1**2 + 512*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 256*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5472*K1*K2*K3 + 1112*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 696*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 680*K2**2*K4 - 1478*K2**2 + 168*K2*K3*K5 + 16*K2*K4*K6 - 684*K3**2 - 154*K4**2 - 20*K5**2 - 2*K6**2 + 1648 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]] |
| If K is slice | False |