| Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,1,1,1,3,2,0,1,2,3,0,0,1,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.626'] |
| 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+58t^5+116t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.626'] |
| 2-strand cable arrow polynomial of the knot is: -32*K1**4 + 128*K1**3*K2*K3 - 128*K1**2*K2**2*K3**2 - 608*K1**2*K2**2 + 272*K1**2*K2 - 512*K1**2*K3**2 - 816*K1**2 + 320*K1*K2**3*K3 + 320*K1*K2*K3**3 + 2016*K1*K2*K3 + 288*K1*K3*K4 + 48*K1*K5*K6 - 176*K2**4 - 736*K2**2*K3**2 - 16*K2**2*K4**2 + 48*K2**2*K4 - 684*K2**2 + 368*K2*K3*K5 + 32*K2*K4*K6 - 160*K3**4 + 16*K3**2*K6 - 696*K3**2 - 76*K4**2 - 88*K5**2 - 36*K6**2 + 906 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.626'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71651', 'vk6.71654', 'vk6.71827', 'vk6.72253', 'vk6.72256', 'vk6.72378', 'vk6.77267', 'vk6.77364', 'vk6.77373', 'vk6.77615', 'vk6.77707', 'vk6.77716', 'vk6.81403', 'vk6.81412', 'vk6.81439', 'vk6.81698', 'vk6.82453', 'vk6.84427', 'vk6.84451', 'vk6.86950', 'vk6.87157', 'vk6.87168', 'vk6.87773', 'vk6.87996', 'vk6.88115', 'vk6.89639'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is -. |
| The reverse -K is |
| The 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 | O1O2O3O4U5O6U3O5U1U6U2U4 |
| R3 orbit | {'O1O2O3O4U5U2O6O5U1U3U6U4', 'O1O2O3O4U5O6U3O5U1U6U2U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U1U3U5U4O6U2O5U6 |
| Gauss code of K* | O1O2O3O4U1U3U5U4O6U2O5U6 |
| Gauss code of -K* | Same |
| 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 -3 0 -1 3 0 1],[ 3 0 2 1 4 2 1],[ 0 -2 0 0 2 0 0],[ 1 -1 0 0 1 1 0],[-3 -4 -2 -1 0 -2 -1],[ 0 -2 0 -1 2 0 1],[-1 -1 0 0 1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 -1 -3],[-3 0 -1 -2 -2 -1 -4],[-1 1 0 0 -1 0 -1],[ 0 2 0 0 0 0 -2],[ 0 2 1 0 0 -1 -2],[ 1 1 0 0 1 0 -1],[ 3 4 1 2 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,1,3,1,2,2,1,4,0,1,0,1,0,0,2,1,2,1] |
| Phi over symmetry | [-3,-1,0,0,1,3,1,1,1,3,2,0,1,2,3,0,0,1,1,1,1] |
| Phi of -K | [-3,-1,0,0,1,3,1,1,1,3,2,0,1,2,3,0,0,1,1,1,1] |
| Phi of K* | [-3,-1,0,0,1,3,1,1,1,3,2,0,1,2,3,0,0,1,1,1,1] |
| Phi of -K* | [-3,-1,0,0,1,3,1,2,2,1,4,0,1,0,1,0,0,2,1,2,1] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^3z+11w^2z+15w |
| Inner characteristic polynomial | t^6+38t^4+56t^2 |
| Outer characteristic polynomial | t^7+58t^5+116t^3 |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -32*K1**4 + 128*K1**3*K2*K3 - 128*K1**2*K2**2*K3**2 - 608*K1**2*K2**2 + 272*K1**2*K2 - 512*K1**2*K3**2 - 816*K1**2 + 320*K1*K2**3*K3 + 320*K1*K2*K3**3 + 2016*K1*K2*K3 + 288*K1*K3*K4 + 48*K1*K5*K6 - 176*K2**4 - 736*K2**2*K3**2 - 16*K2**2*K4**2 + 48*K2**2*K4 - 684*K2**2 + 368*K2*K3*K5 + 32*K2*K4*K6 - 160*K3**4 + 16*K3**2*K6 - 696*K3**2 - 76*K4**2 - 88*K5**2 - 36*K6**2 + 906 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}]] |
| If K is slice | True |