| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,2,2,0,1,1,2,1,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1827'] |
| Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866'] |
| Outer characteristic polynomial of the knot is: t^7+24t^5+43t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1827'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 672*K1**4*K2 - 1504*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 704*K1**3*K3 - 1216*K1**2*K2**2 - 288*K1**2*K2*K4 + 3680*K1**2*K2 - 352*K1**2*K3**2 - 80*K1**2*K4**2 - 2492*K1**2 - 224*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 2656*K1*K2*K3 + 784*K1*K3*K4 + 168*K1*K4*K5 - 24*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 360*K2**2*K4 - 1996*K2**2 + 160*K2*K3*K5 + 32*K2*K4*K6 - 1040*K3**2 - 406*K4**2 - 92*K5**2 - 4*K6**2 + 2084 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1827'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4128', 'vk6.4159', 'vk6.5370', 'vk6.5401', 'vk6.5466', 'vk6.5577', 'vk6.7496', 'vk6.7662', 'vk6.9001', 'vk6.9032', 'vk6.11179', 'vk6.12263', 'vk6.12370', 'vk6.12445', 'vk6.12476', 'vk6.13360', 'vk6.13581', 'vk6.13612', 'vk6.14256', 'vk6.14705', 'vk6.14748', 'vk6.15195', 'vk6.15863', 'vk6.15908', 'vk6.26198', 'vk6.26643', 'vk6.30854', 'vk6.30885', 'vk6.32042', 'vk6.32073', 'vk6.33086', 'vk6.33117', 'vk6.38161', 'vk6.38178', 'vk6.44822', 'vk6.44919', 'vk6.49214', 'vk6.49319', 'vk6.52761', 'vk6.53532'] |
| 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 | O1O2O3U1U3O4U5O6O5U2U4U6 |
| R3 orbit | {'O1O2O3U1U3U4O5O4O6U2U6U5', 'O1O2O3U1U3O4U5O6O5U2U4U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U5U2O6O4U6O5U1U3 |
| Gauss code of K* | O1O2O3U4U1U5O4O5U2O6U3U6 |
| Gauss code of -K* | O1O2O3U4U1O4U2O5O6U5U3U6 |
| 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 -1 1 0 1 1],[ 2 0 2 1 1 2 1],[ 1 -2 0 0 1 1 1],[-1 -1 0 0 0 -1 0],[ 0 -1 -1 0 0 0 0],[-1 -2 -1 1 0 0 1],[-1 -1 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 0 0 0 -1],[-1 -1 0 0 0 -1 -1],[ 0 0 0 0 0 -1 -1],[ 1 1 0 1 1 0 -2],[ 2 2 1 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,0,0,1,0,1,1,1,1,2] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,2,2,0,1,1,2,1,1,1,-1,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,-1,1,1,2,2,0,1,1,2,1,1,1,-1,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,2,1,1,1,1,1,2,2,0,1,-1] |
| Phi of -K* | [-2,-1,0,1,1,1,2,1,1,1,2,1,0,1,1,0,0,0,0,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+16t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+24t^5+43t^3+4t |
| Flat arrow polynomial | -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -192*K1**6 - 128*K1**4*K2**2 + 672*K1**4*K2 - 1504*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 704*K1**3*K3 - 1216*K1**2*K2**2 - 288*K1**2*K2*K4 + 3680*K1**2*K2 - 352*K1**2*K3**2 - 80*K1**2*K4**2 - 2492*K1**2 - 224*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 2656*K1*K2*K3 + 784*K1*K3*K4 + 168*K1*K4*K5 - 24*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 360*K2**2*K4 - 1996*K2**2 + 160*K2*K3*K5 + 32*K2*K4*K6 - 1040*K3**2 - 406*K4**2 - 92*K5**2 - 4*K6**2 + 2084 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |